Research ArticleProving and Generalizing Desarguesrsquo Two-Triangle Theorem in3-Dimensional Projective Space

Dimitrios Kodokostas

Department of Applied Mathematics National Technical University of Athens Zografou Campus 15780 Athens Greece

Correspondence should be addressed to Dimitrios Kodokostas dkodokostasmathntuagr

Received 22 August 2014 Accepted 15 November 2014 Published 18 December 2014

Academic Editor Isaac Pesenson

Copyright copy 2014 Dimitrios Kodokostas This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

With the use of only the incidence axiomsweprove and generalizeDesarguesrsquo two-triangleTheorem in three-dimensional projectivespace considering an arbitrary number of points on each one of the two distinct planes allowing corresponding points on the twoplanes to coincide and three points on any of the planes to be collinear We provide three generalizations and we define the notionsof a generalized line and a triangle-connected plane set of points

1 The Problem in Perspective

Perhaps the most important proposition deduced from theaxioms of incidence in projective geometry for the projectivespace is Desarguesrsquo two-triangleTheorem usually stated quiteconcisely as follows

two triangles in space are perspective from a pointif and only if they are perspective from a line

meaning that if we assume a one-to-one correspondenceamong the vertices of the two triangles then the linesjoining the corresponding vertices are concurrent if andonly if the intersections of the lines of the correspondingsides are collinear (Figure 1) According to the ancient Greekmathematician Pappus (3rd century AD) this theorem wasessentially contained in the lost treatise on Porisms of Euclid(3rd century BC) [1 2] but it is nowadays known by the nameof the French mathematician and military engineer GerardDesargues (1593ndash1662) who published it in 1639 Actuallyonly half of it is called Desarguesrsquo Theorem (perspectivityfrom a point implies perspectivity from a line) whereas theother half is called converse of Desarguesrsquo Theorem Allprinted copies of Desarguesrsquo treatise were lost but fortunatelyDesarguesrsquo contemporary French mathematician Phillipe deLa Hire (1640ndash1718) made a manuscript copy of it which wasdiscovered again some 200 years later [3]

There exist a few hidden assumptions in the abovepretty and compact statement of the theorem Namely all

mentioned lines and intersection points are assumed toexist and the three points on each plane are assumed tobe noncollinear It seems that in bibliography there existvery few treatments of the theorem paying attention to theseassumptions like Hodge and Pedoersquos [4] and Pogorelovrsquos [5]The purpose of this paper is to deal with these assumptionsIt is also worth mentioning that Desargues was interested intriangles in space formed by intersecting two distinct planeswith three lines going through a common point Remarkablyalthough the theorem remains true even when the two planesof the projective space coincide it is not at all evident that itholds whenever we work solely in a general projective planedisregarding a larger surrounding projective space ActuallyDesarguesrsquoTheoremdoes not hold in the projective geometryof a general projective plane defined by the usual incidenceaxioms as Hilbert has shown [6] unless if the theorem itselfor a statement that implies it is assumed as an axiom Sucha set of axioms was given by Bachmann [7] and a proofof Desarguesrsquo Theorem can be found in [8] based on thefollowing Pappusrsquo Theorem of Euclidean Geometry beingconsidered as an axiom

if the six vertices of a hexagon lie alternatively ontwo lines then the three points of intersection ofpairs of opposite sides are collinear

The general projective planes came to existence in anattempt to construct via incidence axioms planes in which

Hindawi Publishing CorporationGeometryVolume 2014 Article ID 276108 7 pageshttpdxdoiorg1011552014276108

2 Geometry

(1)(2)

(3)p

O

(1998400)(2998400)

(3998400) p998400

Figure 1 Desarguesrsquo 2-triangle Theorem in space and its converse

Desarguesrsquo Theorem does or rather does not hold They aredefined as collections of two kinds of objects called points andlines satisfying at least the following three incidence axioms

(i) any two distinct points are incident with exactly oneline

(ii) any two distinct lines are incident with exactly onepoint

(iii) there exist four points of which no three are collinear

where incidence is used as a neutral word meaning that thepoint belongs or lies on the line and also meaning that theline passes through or contains the point Collinearity hasthe usual meaning Some projective planes have only finitelymany points and lines whereas some others have infinitelymany Desarguesrsquo Theorem fails to hold for many of themboth finite and infinite [9ndash12] Hilbert [6] Hall [13] andothers have proved a plane is Desarguesian (for such planesDesarguesrsquoTheoremholds) if and only if it can be constructedalgebraically from the vector spaces1198703 over the division rings119870 where the points of these planes are just the lines of thevector space 1198703 and the lines of the planes are the subspacesof 1198703 spanned by two linearly independent vectors of it Theusual real projective plane comes by this construction fromR3

Nevertheless for projective spaces of dimension at leastthree which are defined similar to the projective planes eitherby a set of incidence axioms or by algebraic constructionsDesarguesrsquo Theorem is always true [4] These higher dimen-sional projective spaces contain 2-dimensional projectiveplanes and it is easily seen that DesarguesrsquoTheorem holds forall such projective planes too

We are going to prove and generalize DesarguesrsquoTheoremworking in a three-dimensional projective space and consid-ering an arbitrary number of points on each one of the twodistinct planes allowing some of the lines and intersectionpoints mentioned in the above statement of the theoremto disappear For this matter we will allow some of thegiven points on one of the planes to coincide with theircorresponding points on the other plane Also although wewill always be assuming that the given points are distinct on

each plane we will drop the restriction that any three of themare noncollinear All our proofs will be based directly on theaxioms of projective geometry in space and their immediatecorollaries

The projective space we will be working on consists ofthree kinds of objects called points lines and planes forwhich the following incidence axioms hold [5]

(i) there exists exactly one line through any two distinctpoints

(ii) any two distinct lines on the same plane have a uniquecommon point

(iii) there exist at least three points on each line(iv) there exist three points not on the same line(v) there exist four points not on the same plane(vi) there exists exactly one plane through any three

distinct noncollinear points there exists at least onepoint on each plane

(vii) if two distinct points are on a plane then all points inthe line containing the given points are on the sameplane

(viii) if two planes have a common point then they have atleast one more common point

Some immediate corollaries are the following

(i) a plane and a line not on the plane always have aunique common point

(ii) two distinct planes intersect in a common line

A number of given points will be called collinear wheneversome line contains all of them (if some points coincide theremay bemore than one such line) Similarly a number of givenlines will be called concurrent whenever there exists at leastone point lying on all of themWewill say a given set of pointson a plane is in general position whenever no three collinearpoints exist among them A usual triangle or just a trianglewill be a triad of lines defined by three noncollinear points

2 A Generalized Desarguesrsquo Theoremfor Points in General Position AllowingCorresponding Points to Coincide

Figure 2(a) makes it clear that if we wish to generalize tolarger numbers of points the converse ofDesarguesrsquoTheoremwe have to assume distinct given points on each planeSimilarly Figure 2(b) says that we should not allow somethree collinear points on a plane whenever some of the restcoincide with their corresponding points on the other planeNevertheless there do exist generalizations if we allow onlyone of these two bothersome situations to happen as we showbelow

So let (1) (2) be given points on a plane 119901 and(11015840) (21015840) some corresponding points on another plane

1199011015840 allowing for the possibility that some corresponding

points coincide Instead of ignoring pairs of coincidingcorresponding points we choose to use generalized lines

Geometry 3

(4)(3)

p

(1998400)

(3998400)

(2998400)

(4998400)

p998400

(1)(2)

(a)

p

(1) (2) (3)

(4)(4998400)

(3998400)

(2998400)

(1998400)

p998400

(b)

Figure 2 No matter what the meaning of the line (1)(2) in (a) and the line (4)(41015840) in (b) is the lines (119894)(1198941015840) cannot all be concurrent

(1) (2)

(3)(4)

O

p

(1998400) (4998400)

(2998400)

(3998400)p998400

120598

Figure 3 Desarguesrsquo Theorem for 4 given points on each plane

defining a generalized line (119894)(1198941015840) to be the line through (119894)and (1198941015840) whenever (119894) (1198941015840) are distinct otherwise to be theset of all lines through (119894) as this seems most natural It isalso convenient to say that (119894)(119894) contains or goes through allpoints in our projective space Our first generalization is thefollowing (Figure 3)

Proposition 1 Let 119901 1199011015840 be two distinct planes (1) (2) (119899) (119899 isin N) distinct points on 119901 and (11015840) (21015840) (1198991015840)distinct points on 1199011015840 both sets of points in general position

(A) If all generalized lines (119894)(1198941015840) go through a commonpoint then all the intersections of the pairs of lines (119894)(119895) (1198941015840)(1198951015840)for 119894 = 119895 are nonempty and lie on a common line (the commonline of 119901 and 1199011015840)

(B) If all the intersections of the pairs of lines (119894)(119895) (1198941015840)(1198951015840)for 119894 = 119895 are nonempty then all generalized lines (119894)(1198941015840) gothrough a common point

Proof For 119899 = 1 part (A) holds vacuously For 119899 ge 2 notethat for arbitrary distinct indices 119894 119895 the generalized lines(119894)(1198941015840) and (119895)(1198951015840) share some common point thus the lines

(119894)(119895) (1198941015840)(1198951015840) should lie on some common plane (Figure 4

provides a quick pictorial proof) and so they should have anonempty intersection But the first of these lines belongsto 119901 whereas the second belongs to 1199011015840 which forces theirintersection to lie on 120598 the common line of the distinct planes119901 1199011015840 Since this happens for all indices 119894 = 119895 we are finished

with part (A)The proof of part (B) is a bit longer For 119899 = 1 the

proposition holds vacuously For 119899 = 2 its truth is immediatewhenever at least one of the generalized lines (1)(11015840) (2)(21015840)is not a line whereas if both of them are lines they eithercoincide (both with the common line of 119901 and 1199011015840) and theresults hold trivially or else they are distinct in which casethe four points (1) (11015840) (2) (21015840) do not lie on the same lineBut then the lines (1)(2) (11015840)(21015840) are also distinct and astheir intersection is nonempty by assumption they definea plane on which lie the points (1) (11015840) (2) (21015840) forcing thelines (1)(11015840) (2)(21015840) to lie on this plane too and thus to havenonempty intersection (a single point since these lines do notcoincide)

For 119899 = 3 and assuming that at least one of the generalizedlines (1)(11015840) (2)(21015840) (3)(31015840) is not a line say (3)(31015840) the resultcomes from the truth of case 119899 = 2 since (1)(11015840) (2)(21015840)have to share some common point which of course lies onthe generalized line (3)(31015840) as well So assume that all threeof these generalized lines are lines (Figure 1) which is theclassical assumption for the converse of DesarguesrsquoTheorem

Observe that at least one of the points (11015840) (21015840) (31015840) say(11015840) does not lie on 119901 otherwise the plane 1199011015840 defined by

these three noncollinear points would coincide with 119901 whichcannot happenThen the line (11015840)(21015840) does not lie on 119901 eitherand so it is distinct from the line (1)(2) which lies on 119901and since (11015840)(21015840) (1)(21015840) have a nonempty intersection theydefine a plane say 119901

12 The line (3)(31015840) cannot lie on 119901

12for

if it did the three noncollinear points (1) (2) (3) would lieon 11990112making it coincide with 119901 and then point (11015840) of 119901

12

would be a point of 119901 a contradiction So the line (3)(31015840) hasa unique common point119874with 119901

12 And we show below that

(3)(31015840) has a unique intersection point with either of lines

4 Geometry

(i)

(i) (i) (i) (i)

(j)

(j)(j) (j) (j)

(i998400) (i998400) (i998400) (i998400)

(i998400)

(j998400)(j998400) (j998400)

(j998400)

(j998400)

Figure 4 Whenever the generalized lines (119894)(1198941015840) (119895)(1198951015840) share a common point the lines (119894)(119895) (1198941015840)(1198951015840) lie on some common plane

(1)(11015840) (2)(2

1015840) of 119901

12 Therefore (1)(11015840) (2)(21015840) (3)(31015840) have

to go through a common point 119874 as wanted Now for theclaim about the intersections of (3)(31015840) with (1)(11015840) (21015840)(21015840)the lines (1)(3) (11015840)(31015840) are distinct otherwise the lines(1)(11015840) (3)(3

1015840) would coincide forcing (3)(31015840) to lie on 119901

12

which it cannot do Then since by hypothesis (1)(3) (11015840)(31015840)intersect nonvacuously they define a plane on which all fourpoints (1) (3) (11015840) (31015840) lie making the lines (1)(11015840) (3)(31015840) lieon it and so have a nonempty intersection This intersectionis actually a unique point otherwise the lines (1)(11015840) (3)(31015840)would coincide forcing (3)(31015840) to lie on 119901

12 a contradiction

Quite similarly (3)(31015840) intersects (21015840)(21015840) in a unique point aswanted

For 119899 ge 4 there exist at least 119899 minus 2 ge 2 lines among the(119894)(1198941015840)rsquos for if not at least three (119894)rsquos would coincide with the

corresponding (1198941015840)rsquos and all these (119894) would be collinear lyingon the common line of 119901 and 1199011015840 contrary to the propositionrsquosassumption

If there exist exactly two lines among the (119894)(1198941015840)rsquos (whichcan happen only for 119899 = 4) say (1)(11015840) and (2)(21015840) thenfor 119894 gt 2 the points (119894) and (1198941015840) coincide and lie on thecommon line 120598 of 119901 1199011015840 Now the lines (1)(2) (11015840)(21015840) cannotcoincide otherwise they would lie on both 119901 1199011015840 and sothey would coincide with 120598 thus all points (119894) would lie on120598 contrary to our original assumption Since (1)(2) (11015840)(21015840)are distinct and by assumption intersect nonvacuously theydefine a plane on which lie all four points (1) (11015840) (2) (21015840)So both lines (1)(11015840) (2)(21015840) lie on this plane as well and theymust have a nonempty intersection a single point Of courseall generalized lines (119894)(1198941015840) go through this point as well

Finally if there exist at least three lines among the (119894)(1198941015840)rsquossay (1)(11015840) (119896)(1198961015840) with 119896 ge 3 then according to theproof of case 119899 = 3 for any 119895 3 le 119895 le 119896 the lines(1)(11015840) (2)(2

1015840) and (119895)(1198951015840) are distinct and intersect each

other in a single point which makes all three of them gothrough a common point say 119874

119895 Since 119874

119895is the unique

common point of the distinct lines (see scholium below)(1)(11015840) (2)(2

1015840) for all indices 119895 all these points coincide and

the lines (1)(11015840) (119896)(1198961015840) go through this common pointwhich of course is also a point of any generalized (119894)(1198941015840) aswanted finishing the proof of the proposition

Remark 2 The above proof reveals that with the classicalhypothesis for the converse of Desarguesrsquo Theorem the threelines (1)(11015840) (2)(21015840) (3)(31015840) are distinct and the three planesdefined by two of them at a time are distinct a result to beused in the next section Indeed with the assumptions madein the proof line (3)(31015840) is distinct from the other two since

(1)

(2)

(3) (4)

p

p998400

(1998400)

(2998400)(4998400)

(3998400)

Figure 5 If (1)(11015840) (2)(21015840) (3)(31015840) are concurrent and (2)(21015840)(3)(31015840) (4)(4

1015840) are concurrent then all (1)(11015840) (2)(21015840) (3)(31015840)

(4)(41015840) are concurrent

it does not lie on the plane 11990112

defined by them whereas if(1)(11015840) (2)(2

1015840) were not distinct the points (1) (11015840) (2) (21015840)

would lie on the same line and this line would be of course(1)(2) forcing (11015840) to lie on (1)(2) and therefore on 119901 acontradiction And any two of the planes defined by two of(1)(11015840) (2)(2

1015840) (3)(3

1015840) have to be distinct since otherwise all

points (1) (2) (3) (11015840) (21015840) (31015840) would lie on the same planeforcing 119901 and 1199011015840 to coincide which cannot be

3 A Generalization for the Converse ofDesarguesrsquo Theorem for Triangle-ConnectedGiven Points

The assumptions postulated in part (B) of Proposition 1 canbe considerably relaxed For example there is no reason forassuming that all lines (119894)(119895) intersect with their correspond-ing lines (1198941015840)(1198951015840) since amuch smaller number of lines amongthe (119894)(119895)rsquos with the same property force all the rest to behavesimilarly

This is made clear in Figure 5 where the lines of the sidesof triangle (1)(2)(3) intersect their corresponding lines oftriangle (11015840)(21015840)(31015840) making the lines (1)(11015840) (2)(21015840) (3)(31015840)go through the same point 119874

1 and similarly the lines of the

sides of triangle (2)(3)(4) intersect their corresponding linesof triangle (21015840)(31015840)(41015840)making the lines (2)(21015840) (3)(31015840) (4)(41015840)go through the same point 119874

2 Since both 119874

1 1198742coin-

cide with the intersection point of (1)(11015840) and (2)(21015840) theyhave to coincide with each other thus making all lines(1)(11015840) (2)(2

1015840) (3)(3

1015840) (4)(4

1015840) go through a common point

Then by part (A) of Proposition 1 lines (1)(4) (11015840)(41015840) haveto intersect as well Of course all the intersections of the pairsof lines (119894)(119895) (1198941015840)(1198951015840) have to lie on the common line of 119901 and1199011015840

Geometry 5

(1)(3)

(2)

(n)

(n minus 1)

(n minus 2)

(a)

(1)

(2)

(3)

(4)

(5)

(n)

(n minus 1)

(n minus 2)

(b)

Figure 6 (a) A triangle-connection set for the points (1) (2) (119899) (b) 119899 minus 4 in number lines can never be all the triangle-connection linesfor 119899 points no three of which are collinear

Also note that the points (1) (2) (3) and (4) need not benoncollinear by three in the argument above we only neededthe existence of the triangles (1)(2)(3) and (2)(3)(4) and theircorresponding ones in 1199011015840 Finally note that if we expand thegiven sets of points with some more points (5) = (51015840) (6) =(61015840) all in the common line of 119901 and 1199011015840 then trivially all

lines (1)(11015840) (2)(21015840) continue to share a common pointThis argument extends similarly for any 119899 ge 3 given

points (119894) and (1198941015840) on the two planes All we need is toassume that all points (119894) that do not coincide with theircorresponding points (1198941015840) are similarly triangle-connected onthe two planes The meaning of this should be rather clear

A triangle-connection sequence in 119878 or for simplicity justtriangle-connection sequence for two points 119860 119861 of a givenset 119878 will be any finite sequence of triangles formed by linesthrough the points of 119878 so that any two consecutive trianglesshare a common side and 119860 belongs to the first triangle ofthis sequence whereas 119861 belongs to the last one recall fromSection 1 that a triangle is a set of three lines A triangle-connection set for 119878 will be a set of triangle-connectionsequences of points of 119878 that contains exactly one suchsequence for each pair of points of 119878 Of coursewhenever sucha set exists for 119878 it is not necessarily unique The triangles inthe sequences of such a set will be called connection trianglesfor 119878 and the lines of the sides connection lines Two givensets of points (1) (119899) and (11015840) (1198991015840) on two planes119901 1199011015840where (119894) corresponds to (1198941015840) will be called similarly triangle-connected whenever there exists a triangle-connection set forthe (119894)rsquos which becomes a triangle-connection set for the (1198941015840)rsquosby replacing all vertices of all triangles in the sequences of thisset by their corresponding points on the other plane

For example a triangle-connection set for the set 119878 =(1) (2) (119899) in Figure 6(a) is 119878

119894119895| 119894 le 119895 + 2 where

119878119894119895= ((119894)(119894+1)(119894+2) (119894+1)(119894+2)(119894+3) (119895minus2)(119895minus1)(119895)) and

triangles are denoted by the three names of their verticesThetriangle-connection lines in this set are (1)(2) (2)(3) (119899minus1)(119899) and (1)(3) (2)(4) (119899 minus 2)(119899) that is 2119899 minus 3 lines in

total It is interesting to note by the way that the minimumnumber of connection lines in a triangle-connection set oflines for 119899 given points no three of which are collinear is 2119899minus3Figure 6(b) hints at the reason a rigorous proof is easy tocome by

Now for two similarly triangle-connected sets of points(119894) and (1198941015840) on two planes 119901 and 1199011015840 and assuming (1)(2)(3)to be a connection triangle on one of the planes weknow by the remark after Proposition 1 that the lines(1)(11015840) (2)(2

1015840) (3)(3

1015840) are distinct and that all three intersect

in a single point 119874 It is quite trivial to show and we do so inthe next paragraph that every line (119894)(1198941015840) intersects all threelines (1)(11015840) (2)(21015840) (3)(31015840) nonvacuously for all 119894 Since for119894 = 1 the line (119894)(1198941015840) does not belong to at least one of thedistinct planes defined by (1)(11015840) (2)(21015840) and (1)(11015840) (3)(31015840)we get that (119894)(1198941015840) has a unique point with at least one ofthe planes In other words (119894)(1198941015840) either intersects (1)(11015840) and(2)(21015840) of the first of these planes on their common point119874 or

else intersects (1)(11015840) and (3)(31015840) of the second plane on theircommon point 119874 So all lines (119894)(1198941015840) go through 119874 provingProposition 3 below which is a generalization of the converseof Desarguesrsquo Theorem

About the claim that for all 119894 the line (119894)(1198941015840) intersectsnonvacuously all three lines (1)(11015840) (2)(21015840) (3)(31015840) for (119894) =(1198941015840) this is immediate so consider an 119894 so that (119894)(1198941015840) is a

usual line and choose a triangle-connection sequence for (1)and (119894) For any triangle (119894

1)(1198942)(1198943) in this sequence and its

corresponding (11989410158401)(1198941015840

2)(1198941015840

3) on the other plane the assumptions

for the converse of the classical Desargues Theorem aresatisfied so (see the scholium) the lines (119894

1)(1198941015840

1) (1198942)(1198941015840

2) (1198943)(1198941015840

3)

are distinct and all three intersect in a single point Now fortwo such consecutive triangles (119894

1)(1198942)(1198943) (1198942)(1198943)(1198944) the two

concurrency points are the same since they both coincidewith the unique intersection point of (119894

2)(1198941015840

2) and (119894

3)(1198941015840

3)This

implies that all lines (119895)(1198951015840) for all the vertices (119895) (1198951015840) of alltriangles in these triangle-connection sequences have to gothrough a common point which implies (1)(11015840) and (119894)(1198941015840)

6 Geometry

p

(1) (4)(3)(2)

p998400

(4998400)(3998400)

(2998400)(1998400)

Figure 7The converse of DesarguesrsquoTheorem does not always holdin case all given points are collinear in both planes

have some common point Similarly (119894)(1198941015840) intersects (2)(21015840)and (3)(31015840) as well and we are done

Proposition 3 Let119901 1199011015840 be two distinct planes (1) (2) (119899)distinct points on 119901 and (11015840) (21015840) (1198991015840) distinct points on 1199011015840where (119894) (1198941015840) are considered to be corresponding Assume thatfor the given points on the two planes which do not coincidewith their corresponding points there exist similar triangle-connection sets so that every pair of corresponding triangle-connection lines (119894)(119895) (1198941015840)(1198951015840) has a nonempty intersectionThen all generalized lines (119894)(1198941015840) go through a common pointMoreover all pairs of lines (119894)(119895) (1198941015840)(1198951015840) for 119894 = 119895 intersectnonvacuously (on the common line of 119901 and 1199011015840)

The existence of the triangle-connection sets guaranteesthat 119899 ge 3 and that at least three points on each of theplanes do not coincide with their corresponding points onthe other If exactly two points on each plane do not coincidewith their corresponding points then all but two lines (119894)(119894)are generalized but the remaining two are not guaranteed tohave a nonempty intersection and thus this case cannot beincluded to the proposition

4 Whenever Three or More Collinear PointsAre Allowed but Corresponding Points AreAssumed Distinct

We close with a third generalization of the converse ofDesarguesrsquoTheorem In Proposition 1 the given points on thetwo planes were allowed to coincide with their correspondingpoints but no three given points on any of the planes wereallowed to be collinear Flipping over these assumptions andrestrictions we now forbid any two corresponding pointsto coincide but allow three or more of them on any of theplanes to be collinear Figure 7 reveals that the converse ofDesarguesrsquo Theorem does not always hold then at least itdoes not hold whenever the given points on the planes areall collinear Interestingly enough it does hold whenever thegiven points on one of the planes do not all lie in a single line

Proposition 4 Let 119901 1199011015840 be two distinct planes (1) (2) (119899) 119899 ge 3 distinct points on 119901 and (11015840) (21015840) (1198991015840) distinctpoints on 1199011015840 so that (119894) = (1198941015840) for all 119894 If all intersections(119894)(119895) cap (119894

1015840)(1198951015840) are nonempty and not all of the given points

are collinear on at least one of the given planes then all lines(119894)(1198941015840) go through a common point

Proof For 119899 = 3 this is exactly part (B) of Proposition 1Similarly it is part (B) of Proposition 1whenever 119899 = 4 and nothree collinear points exist among the given ones in any of theplanes For 119899 = 4 and assuming (1) (2) (3) to be collinear on119901 then (4) does not lie on the line of the other three and so(1)(2)(4) (2)(4)(3) are usual nontrivial triangles the points(1) (2) (3) (4) are triangle-connected and our propositionholds because of Proposition 3

Finally for 119899 ge 5 we proceed inductively assuming theresult holds for all 119896 le 119899 by the propositionrsquos assumptionswe can consider (119894) to be not all collinear If all but oneof them are collinear say (1) not on the line of the restthen (1)(2)(3) (1)(3)(4) (1)(119899 minus 1)(1) form a triangle-connection set for the (119894)rsquos and the result follows fromProposition 3

If on the other hand no more than 119899 minus 2 of the (119894)rsquos arecollinear choose two of them say (1) (2) not on the commonline 120598 of 119901 1199011015840 Now observe that the two sets of 119899 minus 1 innumber points (1) (2) (119899minus2) (119899minus1) and (1) (2) (119899minus2) (119899) satisfy the assumptions of the proposition thus byour induction hypothesis the result holds for them In otherwords all lines (2)(21015840) (3)(31015840) (119899 minus 2)(119899 minus 21015840) (119899 minus 1)(119899 minus11015840) go through a common point 119874

1of (1)(11015840) (2)(21015840) and

similarly all lines (2)(21015840) (3)(31015840) (119899 minus 2)(119899 minus 21015840) (119899)(1198991015840)go through a point 119874

2of (1)(11015840) (2)(21015840) But points 119874

1 1198742

coincide because the lines (1)(11015840) (2)(21015840) are distinct if not(1)(11015840) (2)(2

1015840) would coincide with 120598 because they would be

the line (1)(2) of 119901 and at the same time the line (11015840)(21015840) of1199011015840 this would make (1) (2) points of 120598 a contradiction So all

lines (119894)(1198941015840) go through 1198741and we are done

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] H Dorrie 100 Great Problems of ElementaryMathematicsTheirHistory and Solution Dover 1965

[2] W R Knorr The Ancient Tradition of Geometric ProblemsDover Publications New York NY USA 1993

[3] M Kline Mathematics in Western Culture Oxford UniversityPress 1967

[4] W V Hodge and D PedoeMethods of Algebraic Geometry vol1 Cambridge University Press Cambridge UK 1953

[5] A Pogorelov Geometry Mir Publishers Moscow Russia 1987[6] D Hilbert Grundlagen der Geometrie 1899 (translated asThe

Foundations of Geometry and first published in English byTheOpenCourt Publishing Company Chicago 1992 and offered as anebookbyProject Gutenberg 2005 httpswwwgutenbergorg)

[7] I Bachmann Aufbau der Geometrie aus dem SpiegelungsbegriffGrundlehren derMathematischenWissenschaften 96 SpringerBerlin Germany 2nd edition 1973

[8] H S M Coxeter Introduction to Geometry JohnWiley amp SonsInc New York NY USA 2nd edition 1989

Geometry 7

[9] K Levenberg ldquoA class of non-desarguesian plane geometriesrdquoThe American Mathematical Monthly vol 57 pp 381ndash387 1950

[10] F Moulton ldquoA simple non-Desarguesian plane geometryrdquoTransactions of the American Mathematical Society vol 3 pp102ndash195 1901

[11] K Sitaram ldquoA real non-desarguesian planerdquo The AmericanMathematical Monthly vol 70 pp 522ndash525 1963

[12] OVeblen and JHMaclagan-Wedderburn ldquoNon-Desarguesianand non-Pascalian geometriesrdquo Transactions of the AmericanMathematical Society vol 8 no 3 pp 379ndash388 1907

[13] M Hall ldquoProjective planesrdquo Transactions of the AmericanMathematical Society vol 54 pp 229ndash277 1943

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Stochastic AnalysisInternational Journal of

2 Geometry

(1)(2)

(3)p

O

(1998400)(2998400)

(3998400) p998400

Figure 1 Desarguesrsquo 2-triangle Theorem in space and its converse

Desarguesrsquo Theorem does or rather does not hold They aredefined as collections of two kinds of objects called points andlines satisfying at least the following three incidence axioms

(i) any two distinct points are incident with exactly oneline

(ii) any two distinct lines are incident with exactly onepoint

(iii) there exist four points of which no three are collinear

where incidence is used as a neutral word meaning that thepoint belongs or lies on the line and also meaning that theline passes through or contains the point Collinearity hasthe usual meaning Some projective planes have only finitelymany points and lines whereas some others have infinitelymany Desarguesrsquo Theorem fails to hold for many of themboth finite and infinite [9ndash12] Hilbert [6] Hall [13] andothers have proved a plane is Desarguesian (for such planesDesarguesrsquoTheoremholds) if and only if it can be constructedalgebraically from the vector spaces1198703 over the division rings119870 where the points of these planes are just the lines of thevector space 1198703 and the lines of the planes are the subspacesof 1198703 spanned by two linearly independent vectors of it Theusual real projective plane comes by this construction fromR3

Nevertheless for projective spaces of dimension at leastthree which are defined similar to the projective planes eitherby a set of incidence axioms or by algebraic constructionsDesarguesrsquo Theorem is always true [4] These higher dimen-sional projective spaces contain 2-dimensional projectiveplanes and it is easily seen that DesarguesrsquoTheorem holds forall such projective planes too

We are going to prove and generalize DesarguesrsquoTheoremworking in a three-dimensional projective space and consid-ering an arbitrary number of points on each one of the twodistinct planes allowing some of the lines and intersectionpoints mentioned in the above statement of the theoremto disappear For this matter we will allow some of thegiven points on one of the planes to coincide with theircorresponding points on the other plane Also although wewill always be assuming that the given points are distinct on

each plane we will drop the restriction that any three of themare noncollinear All our proofs will be based directly on theaxioms of projective geometry in space and their immediatecorollaries

The projective space we will be working on consists ofthree kinds of objects called points lines and planes forwhich the following incidence axioms hold [5]

(i) there exists exactly one line through any two distinctpoints

(ii) any two distinct lines on the same plane have a uniquecommon point

(iii) there exist at least three points on each line(iv) there exist three points not on the same line(v) there exist four points not on the same plane(vi) there exists exactly one plane through any three

distinct noncollinear points there exists at least onepoint on each plane

(vii) if two distinct points are on a plane then all points inthe line containing the given points are on the sameplane

(viii) if two planes have a common point then they have atleast one more common point

Some immediate corollaries are the following

(i) a plane and a line not on the plane always have aunique common point

(ii) two distinct planes intersect in a common line

A number of given points will be called collinear wheneversome line contains all of them (if some points coincide theremay bemore than one such line) Similarly a number of givenlines will be called concurrent whenever there exists at leastone point lying on all of themWewill say a given set of pointson a plane is in general position whenever no three collinearpoints exist among them A usual triangle or just a trianglewill be a triad of lines defined by three noncollinear points

2 A Generalized Desarguesrsquo Theoremfor Points in General Position AllowingCorresponding Points to Coincide

Figure 2(a) makes it clear that if we wish to generalize tolarger numbers of points the converse ofDesarguesrsquoTheoremwe have to assume distinct given points on each planeSimilarly Figure 2(b) says that we should not allow somethree collinear points on a plane whenever some of the restcoincide with their corresponding points on the other planeNevertheless there do exist generalizations if we allow onlyone of these two bothersome situations to happen as we showbelow

So let (1) (2) be given points on a plane 119901 and(11015840) (21015840) some corresponding points on another plane

1199011015840 allowing for the possibility that some corresponding

points coincide Instead of ignoring pairs of coincidingcorresponding points we choose to use generalized lines

Geometry 3

(4)(3)

p

(1998400)

(3998400)

(2998400)

(4998400)

p998400

(1)(2)

(a)

p

(1) (2) (3)

(4)(4998400)

(3998400)

(2998400)

(1998400)

p998400

(b)

Figure 2 No matter what the meaning of the line (1)(2) in (a) and the line (4)(41015840) in (b) is the lines (119894)(1198941015840) cannot all be concurrent

(1) (2)

(3)(4)

O

p

(1998400) (4998400)

(2998400)

(3998400)p998400

120598

Figure 3 Desarguesrsquo Theorem for 4 given points on each plane

defining a generalized line (119894)(1198941015840) to be the line through (119894)and (1198941015840) whenever (119894) (1198941015840) are distinct otherwise to be theset of all lines through (119894) as this seems most natural It isalso convenient to say that (119894)(119894) contains or goes through allpoints in our projective space Our first generalization is thefollowing (Figure 3)

Proposition 1 Let 119901 1199011015840 be two distinct planes (1) (2) (119899) (119899 isin N) distinct points on 119901 and (11015840) (21015840) (1198991015840)distinct points on 1199011015840 both sets of points in general position

(A) If all generalized lines (119894)(1198941015840) go through a commonpoint then all the intersections of the pairs of lines (119894)(119895) (1198941015840)(1198951015840)for 119894 = 119895 are nonempty and lie on a common line (the commonline of 119901 and 1199011015840)

(B) If all the intersections of the pairs of lines (119894)(119895) (1198941015840)(1198951015840)for 119894 = 119895 are nonempty then all generalized lines (119894)(1198941015840) gothrough a common point

Proof For 119899 = 1 part (A) holds vacuously For 119899 ge 2 notethat for arbitrary distinct indices 119894 119895 the generalized lines(119894)(1198941015840) and (119895)(1198951015840) share some common point thus the lines

(119894)(119895) (1198941015840)(1198951015840) should lie on some common plane (Figure 4

provides a quick pictorial proof) and so they should have anonempty intersection But the first of these lines belongsto 119901 whereas the second belongs to 1199011015840 which forces theirintersection to lie on 120598 the common line of the distinct planes119901 1199011015840 Since this happens for all indices 119894 = 119895 we are finished

with part (A)The proof of part (B) is a bit longer For 119899 = 1 the

proposition holds vacuously For 119899 = 2 its truth is immediatewhenever at least one of the generalized lines (1)(11015840) (2)(21015840)is not a line whereas if both of them are lines they eithercoincide (both with the common line of 119901 and 1199011015840) and theresults hold trivially or else they are distinct in which casethe four points (1) (11015840) (2) (21015840) do not lie on the same lineBut then the lines (1)(2) (11015840)(21015840) are also distinct and astheir intersection is nonempty by assumption they definea plane on which lie the points (1) (11015840) (2) (21015840) forcing thelines (1)(11015840) (2)(21015840) to lie on this plane too and thus to havenonempty intersection (a single point since these lines do notcoincide)

For 119899 = 3 and assuming that at least one of the generalizedlines (1)(11015840) (2)(21015840) (3)(31015840) is not a line say (3)(31015840) the resultcomes from the truth of case 119899 = 2 since (1)(11015840) (2)(21015840)have to share some common point which of course lies onthe generalized line (3)(31015840) as well So assume that all threeof these generalized lines are lines (Figure 1) which is theclassical assumption for the converse of DesarguesrsquoTheorem

Observe that at least one of the points (11015840) (21015840) (31015840) say(11015840) does not lie on 119901 otherwise the plane 1199011015840 defined by

these three noncollinear points would coincide with 119901 whichcannot happenThen the line (11015840)(21015840) does not lie on 119901 eitherand so it is distinct from the line (1)(2) which lies on 119901and since (11015840)(21015840) (1)(21015840) have a nonempty intersection theydefine a plane say 119901

12 The line (3)(31015840) cannot lie on 119901

12for

if it did the three noncollinear points (1) (2) (3) would lieon 11990112making it coincide with 119901 and then point (11015840) of 119901

12

would be a point of 119901 a contradiction So the line (3)(31015840) hasa unique common point119874with 119901

12 And we show below that

(3)(31015840) has a unique intersection point with either of lines

4 Geometry

(i)

(i) (i) (i) (i)

(j)

(j)(j) (j) (j)

(i998400) (i998400) (i998400) (i998400)

(i998400)

(j998400)(j998400) (j998400)

(j998400)

(j998400)

Figure 4 Whenever the generalized lines (119894)(1198941015840) (119895)(1198951015840) share a common point the lines (119894)(119895) (1198941015840)(1198951015840) lie on some common plane

(1)(11015840) (2)(2

1015840) of 119901

12 Therefore (1)(11015840) (2)(21015840) (3)(31015840) have

to go through a common point 119874 as wanted Now for theclaim about the intersections of (3)(31015840) with (1)(11015840) (21015840)(21015840)the lines (1)(3) (11015840)(31015840) are distinct otherwise the lines(1)(11015840) (3)(3

1015840) would coincide forcing (3)(31015840) to lie on 119901

12

which it cannot do Then since by hypothesis (1)(3) (11015840)(31015840)intersect nonvacuously they define a plane on which all fourpoints (1) (3) (11015840) (31015840) lie making the lines (1)(11015840) (3)(31015840) lieon it and so have a nonempty intersection This intersectionis actually a unique point otherwise the lines (1)(11015840) (3)(31015840)would coincide forcing (3)(31015840) to lie on 119901

12 a contradiction

Quite similarly (3)(31015840) intersects (21015840)(21015840) in a unique point aswanted

For 119899 ge 4 there exist at least 119899 minus 2 ge 2 lines among the(119894)(1198941015840)rsquos for if not at least three (119894)rsquos would coincide with the

corresponding (1198941015840)rsquos and all these (119894) would be collinear lyingon the common line of 119901 and 1199011015840 contrary to the propositionrsquosassumption

If there exist exactly two lines among the (119894)(1198941015840)rsquos (whichcan happen only for 119899 = 4) say (1)(11015840) and (2)(21015840) thenfor 119894 gt 2 the points (119894) and (1198941015840) coincide and lie on thecommon line 120598 of 119901 1199011015840 Now the lines (1)(2) (11015840)(21015840) cannotcoincide otherwise they would lie on both 119901 1199011015840 and sothey would coincide with 120598 thus all points (119894) would lie on120598 contrary to our original assumption Since (1)(2) (11015840)(21015840)are distinct and by assumption intersect nonvacuously theydefine a plane on which lie all four points (1) (11015840) (2) (21015840)So both lines (1)(11015840) (2)(21015840) lie on this plane as well and theymust have a nonempty intersection a single point Of courseall generalized lines (119894)(1198941015840) go through this point as well

Finally if there exist at least three lines among the (119894)(1198941015840)rsquossay (1)(11015840) (119896)(1198961015840) with 119896 ge 3 then according to theproof of case 119899 = 3 for any 119895 3 le 119895 le 119896 the lines(1)(11015840) (2)(2

1015840) and (119895)(1198951015840) are distinct and intersect each

other in a single point which makes all three of them gothrough a common point say 119874

119895 Since 119874

119895is the unique

common point of the distinct lines (see scholium below)(1)(11015840) (2)(2

1015840) for all indices 119895 all these points coincide and

the lines (1)(11015840) (119896)(1198961015840) go through this common pointwhich of course is also a point of any generalized (119894)(1198941015840) aswanted finishing the proof of the proposition

Remark 2 The above proof reveals that with the classicalhypothesis for the converse of Desarguesrsquo Theorem the threelines (1)(11015840) (2)(21015840) (3)(31015840) are distinct and the three planesdefined by two of them at a time are distinct a result to beused in the next section Indeed with the assumptions madein the proof line (3)(31015840) is distinct from the other two since

(1)

(2)

(3) (4)

p

p998400

(1998400)

(2998400)(4998400)

(3998400)

Figure 5 If (1)(11015840) (2)(21015840) (3)(31015840) are concurrent and (2)(21015840)(3)(31015840) (4)(4

1015840) are concurrent then all (1)(11015840) (2)(21015840) (3)(31015840)

(4)(41015840) are concurrent

it does not lie on the plane 11990112

defined by them whereas if(1)(11015840) (2)(2

1015840) were not distinct the points (1) (11015840) (2) (21015840)

would lie on the same line and this line would be of course(1)(2) forcing (11015840) to lie on (1)(2) and therefore on 119901 acontradiction And any two of the planes defined by two of(1)(11015840) (2)(2

1015840) (3)(3

1015840) have to be distinct since otherwise all

points (1) (2) (3) (11015840) (21015840) (31015840) would lie on the same planeforcing 119901 and 1199011015840 to coincide which cannot be

3 A Generalization for the Converse ofDesarguesrsquo Theorem for Triangle-ConnectedGiven Points

The assumptions postulated in part (B) of Proposition 1 canbe considerably relaxed For example there is no reason forassuming that all lines (119894)(119895) intersect with their correspond-ing lines (1198941015840)(1198951015840) since amuch smaller number of lines amongthe (119894)(119895)rsquos with the same property force all the rest to behavesimilarly

This is made clear in Figure 5 where the lines of the sidesof triangle (1)(2)(3) intersect their corresponding lines oftriangle (11015840)(21015840)(31015840) making the lines (1)(11015840) (2)(21015840) (3)(31015840)go through the same point 119874

1 and similarly the lines of the

sides of triangle (2)(3)(4) intersect their corresponding linesof triangle (21015840)(31015840)(41015840)making the lines (2)(21015840) (3)(31015840) (4)(41015840)go through the same point 119874

2 Since both 119874

1 1198742coin-

cide with the intersection point of (1)(11015840) and (2)(21015840) theyhave to coincide with each other thus making all lines(1)(11015840) (2)(2

1015840) (3)(3

1015840) (4)(4

1015840) go through a common point

Then by part (A) of Proposition 1 lines (1)(4) (11015840)(41015840) haveto intersect as well Of course all the intersections of the pairsof lines (119894)(119895) (1198941015840)(1198951015840) have to lie on the common line of 119901 and1199011015840

Geometry 5

(1)(3)

(2)

(n)

(n minus 1)

(n minus 2)

(a)

(1)

(2)

(3)

(4)

(5)

(n)

(n minus 1)

(n minus 2)

(b)

Figure 6 (a) A triangle-connection set for the points (1) (2) (119899) (b) 119899 minus 4 in number lines can never be all the triangle-connection linesfor 119899 points no three of which are collinear

Also note that the points (1) (2) (3) and (4) need not benoncollinear by three in the argument above we only neededthe existence of the triangles (1)(2)(3) and (2)(3)(4) and theircorresponding ones in 1199011015840 Finally note that if we expand thegiven sets of points with some more points (5) = (51015840) (6) =(61015840) all in the common line of 119901 and 1199011015840 then trivially all

lines (1)(11015840) (2)(21015840) continue to share a common pointThis argument extends similarly for any 119899 ge 3 given

points (119894) and (1198941015840) on the two planes All we need is toassume that all points (119894) that do not coincide with theircorresponding points (1198941015840) are similarly triangle-connected onthe two planes The meaning of this should be rather clear

A triangle-connection sequence in 119878 or for simplicity justtriangle-connection sequence for two points 119860 119861 of a givenset 119878 will be any finite sequence of triangles formed by linesthrough the points of 119878 so that any two consecutive trianglesshare a common side and 119860 belongs to the first triangle ofthis sequence whereas 119861 belongs to the last one recall fromSection 1 that a triangle is a set of three lines A triangle-connection set for 119878 will be a set of triangle-connectionsequences of points of 119878 that contains exactly one suchsequence for each pair of points of 119878 Of coursewhenever sucha set exists for 119878 it is not necessarily unique The triangles inthe sequences of such a set will be called connection trianglesfor 119878 and the lines of the sides connection lines Two givensets of points (1) (119899) and (11015840) (1198991015840) on two planes119901 1199011015840where (119894) corresponds to (1198941015840) will be called similarly triangle-connected whenever there exists a triangle-connection set forthe (119894)rsquos which becomes a triangle-connection set for the (1198941015840)rsquosby replacing all vertices of all triangles in the sequences of thisset by their corresponding points on the other plane

For example a triangle-connection set for the set 119878 =(1) (2) (119899) in Figure 6(a) is 119878

119894119895| 119894 le 119895 + 2 where

119878119894119895= ((119894)(119894+1)(119894+2) (119894+1)(119894+2)(119894+3) (119895minus2)(119895minus1)(119895)) and

triangles are denoted by the three names of their verticesThetriangle-connection lines in this set are (1)(2) (2)(3) (119899minus1)(119899) and (1)(3) (2)(4) (119899 minus 2)(119899) that is 2119899 minus 3 lines in

total It is interesting to note by the way that the minimumnumber of connection lines in a triangle-connection set oflines for 119899 given points no three of which are collinear is 2119899minus3Figure 6(b) hints at the reason a rigorous proof is easy tocome by

Now for two similarly triangle-connected sets of points(119894) and (1198941015840) on two planes 119901 and 1199011015840 and assuming (1)(2)(3)to be a connection triangle on one of the planes weknow by the remark after Proposition 1 that the lines(1)(11015840) (2)(2

1015840) (3)(3

1015840) are distinct and that all three intersect

in a single point 119874 It is quite trivial to show and we do so inthe next paragraph that every line (119894)(1198941015840) intersects all threelines (1)(11015840) (2)(21015840) (3)(31015840) nonvacuously for all 119894 Since for119894 = 1 the line (119894)(1198941015840) does not belong to at least one of thedistinct planes defined by (1)(11015840) (2)(21015840) and (1)(11015840) (3)(31015840)we get that (119894)(1198941015840) has a unique point with at least one ofthe planes In other words (119894)(1198941015840) either intersects (1)(11015840) and(2)(21015840) of the first of these planes on their common point119874 or

else intersects (1)(11015840) and (3)(31015840) of the second plane on theircommon point 119874 So all lines (119894)(1198941015840) go through 119874 provingProposition 3 below which is a generalization of the converseof Desarguesrsquo Theorem

About the claim that for all 119894 the line (119894)(1198941015840) intersectsnonvacuously all three lines (1)(11015840) (2)(21015840) (3)(31015840) for (119894) =(1198941015840) this is immediate so consider an 119894 so that (119894)(1198941015840) is a

usual line and choose a triangle-connection sequence for (1)and (119894) For any triangle (119894

1)(1198942)(1198943) in this sequence and its

corresponding (11989410158401)(1198941015840

2)(1198941015840

3) on the other plane the assumptions

for the converse of the classical Desargues Theorem aresatisfied so (see the scholium) the lines (119894

1)(1198941015840

1) (1198942)(1198941015840

2) (1198943)(1198941015840

3)

are distinct and all three intersect in a single point Now fortwo such consecutive triangles (119894

1)(1198942)(1198943) (1198942)(1198943)(1198944) the two

concurrency points are the same since they both coincidewith the unique intersection point of (119894

2)(1198941015840

2) and (119894

3)(1198941015840

3)This

implies that all lines (119895)(1198951015840) for all the vertices (119895) (1198951015840) of alltriangles in these triangle-connection sequences have to gothrough a common point which implies (1)(11015840) and (119894)(1198941015840)

6 Geometry

p

(1) (4)(3)(2)

p998400

(4998400)(3998400)

(2998400)(1998400)

Figure 7The converse of DesarguesrsquoTheorem does not always holdin case all given points are collinear in both planes

have some common point Similarly (119894)(1198941015840) intersects (2)(21015840)and (3)(31015840) as well and we are done

Proposition 3 Let119901 1199011015840 be two distinct planes (1) (2) (119899)distinct points on 119901 and (11015840) (21015840) (1198991015840) distinct points on 1199011015840where (119894) (1198941015840) are considered to be corresponding Assume thatfor the given points on the two planes which do not coincidewith their corresponding points there exist similar triangle-connection sets so that every pair of corresponding triangle-connection lines (119894)(119895) (1198941015840)(1198951015840) has a nonempty intersectionThen all generalized lines (119894)(1198941015840) go through a common pointMoreover all pairs of lines (119894)(119895) (1198941015840)(1198951015840) for 119894 = 119895 intersectnonvacuously (on the common line of 119901 and 1199011015840)

The existence of the triangle-connection sets guaranteesthat 119899 ge 3 and that at least three points on each of theplanes do not coincide with their corresponding points onthe other If exactly two points on each plane do not coincidewith their corresponding points then all but two lines (119894)(119894)are generalized but the remaining two are not guaranteed tohave a nonempty intersection and thus this case cannot beincluded to the proposition

4 Whenever Three or More Collinear PointsAre Allowed but Corresponding Points AreAssumed Distinct

We close with a third generalization of the converse ofDesarguesrsquoTheorem In Proposition 1 the given points on thetwo planes were allowed to coincide with their correspondingpoints but no three given points on any of the planes wereallowed to be collinear Flipping over these assumptions andrestrictions we now forbid any two corresponding pointsto coincide but allow three or more of them on any of theplanes to be collinear Figure 7 reveals that the converse ofDesarguesrsquo Theorem does not always hold then at least itdoes not hold whenever the given points on the planes areall collinear Interestingly enough it does hold whenever thegiven points on one of the planes do not all lie in a single line

Proposition 4 Let 119901 1199011015840 be two distinct planes (1) (2) (119899) 119899 ge 3 distinct points on 119901 and (11015840) (21015840) (1198991015840) distinctpoints on 1199011015840 so that (119894) = (1198941015840) for all 119894 If all intersections(119894)(119895) cap (119894

1015840)(1198951015840) are nonempty and not all of the given points

are collinear on at least one of the given planes then all lines(119894)(1198941015840) go through a common point

Proof For 119899 = 3 this is exactly part (B) of Proposition 1Similarly it is part (B) of Proposition 1whenever 119899 = 4 and nothree collinear points exist among the given ones in any of theplanes For 119899 = 4 and assuming (1) (2) (3) to be collinear on119901 then (4) does not lie on the line of the other three and so(1)(2)(4) (2)(4)(3) are usual nontrivial triangles the points(1) (2) (3) (4) are triangle-connected and our propositionholds because of Proposition 3

Finally for 119899 ge 5 we proceed inductively assuming theresult holds for all 119896 le 119899 by the propositionrsquos assumptionswe can consider (119894) to be not all collinear If all but oneof them are collinear say (1) not on the line of the restthen (1)(2)(3) (1)(3)(4) (1)(119899 minus 1)(1) form a triangle-connection set for the (119894)rsquos and the result follows fromProposition 3

If on the other hand no more than 119899 minus 2 of the (119894)rsquos arecollinear choose two of them say (1) (2) not on the commonline 120598 of 119901 1199011015840 Now observe that the two sets of 119899 minus 1 innumber points (1) (2) (119899minus2) (119899minus1) and (1) (2) (119899minus2) (119899) satisfy the assumptions of the proposition thus byour induction hypothesis the result holds for them In otherwords all lines (2)(21015840) (3)(31015840) (119899 minus 2)(119899 minus 21015840) (119899 minus 1)(119899 minus11015840) go through a common point 119874

1of (1)(11015840) (2)(21015840) and

similarly all lines (2)(21015840) (3)(31015840) (119899 minus 2)(119899 minus 21015840) (119899)(1198991015840)go through a point 119874

2of (1)(11015840) (2)(21015840) But points 119874

1 1198742

coincide because the lines (1)(11015840) (2)(21015840) are distinct if not(1)(11015840) (2)(2

1015840) would coincide with 120598 because they would be

the line (1)(2) of 119901 and at the same time the line (11015840)(21015840) of1199011015840 this would make (1) (2) points of 120598 a contradiction So all

lines (119894)(1198941015840) go through 1198741and we are done

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] H Dorrie 100 Great Problems of ElementaryMathematicsTheirHistory and Solution Dover 1965

[2] W R Knorr The Ancient Tradition of Geometric ProblemsDover Publications New York NY USA 1993

[3] M Kline Mathematics in Western Culture Oxford UniversityPress 1967

[4] W V Hodge and D PedoeMethods of Algebraic Geometry vol1 Cambridge University Press Cambridge UK 1953

[5] A Pogorelov Geometry Mir Publishers Moscow Russia 1987[6] D Hilbert Grundlagen der Geometrie 1899 (translated asThe

Foundations of Geometry and first published in English byTheOpenCourt Publishing Company Chicago 1992 and offered as anebookbyProject Gutenberg 2005 httpswwwgutenbergorg)

[7] I Bachmann Aufbau der Geometrie aus dem SpiegelungsbegriffGrundlehren derMathematischenWissenschaften 96 SpringerBerlin Germany 2nd edition 1973

[8] H S M Coxeter Introduction to Geometry JohnWiley amp SonsInc New York NY USA 2nd edition 1989

Geometry 7

[9] K Levenberg ldquoA class of non-desarguesian plane geometriesrdquoThe American Mathematical Monthly vol 57 pp 381ndash387 1950

[10] F Moulton ldquoA simple non-Desarguesian plane geometryrdquoTransactions of the American Mathematical Society vol 3 pp102ndash195 1901

[11] K Sitaram ldquoA real non-desarguesian planerdquo The AmericanMathematical Monthly vol 70 pp 522ndash525 1963

[12] OVeblen and JHMaclagan-Wedderburn ldquoNon-Desarguesianand non-Pascalian geometriesrdquo Transactions of the AmericanMathematical Society vol 8 no 3 pp 379ndash388 1907

[13] M Hall ldquoProjective planesrdquo Transactions of the AmericanMathematical Society vol 54 pp 229ndash277 1943

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Geometry 3

(4)(3)

p

(1998400)

(3998400)

(2998400)

(4998400)

p998400

(1)(2)

(a)

p

(1) (2) (3)

(4)(4998400)

(3998400)

(2998400)

(1998400)

p998400

(b)

Figure 2 No matter what the meaning of the line (1)(2) in (a) and the line (4)(41015840) in (b) is the lines (119894)(1198941015840) cannot all be concurrent

(1) (2)

(3)(4)

O

p

(1998400) (4998400)

(2998400)

(3998400)p998400

120598

Figure 3 Desarguesrsquo Theorem for 4 given points on each plane

defining a generalized line (119894)(1198941015840) to be the line through (119894)and (1198941015840) whenever (119894) (1198941015840) are distinct otherwise to be theset of all lines through (119894) as this seems most natural It isalso convenient to say that (119894)(119894) contains or goes through allpoints in our projective space Our first generalization is thefollowing (Figure 3)

Proposition 1 Let 119901 1199011015840 be two distinct planes (1) (2) (119899) (119899 isin N) distinct points on 119901 and (11015840) (21015840) (1198991015840)distinct points on 1199011015840 both sets of points in general position

(A) If all generalized lines (119894)(1198941015840) go through a commonpoint then all the intersections of the pairs of lines (119894)(119895) (1198941015840)(1198951015840)for 119894 = 119895 are nonempty and lie on a common line (the commonline of 119901 and 1199011015840)

(B) If all the intersections of the pairs of lines (119894)(119895) (1198941015840)(1198951015840)for 119894 = 119895 are nonempty then all generalized lines (119894)(1198941015840) gothrough a common point

Proof For 119899 = 1 part (A) holds vacuously For 119899 ge 2 notethat for arbitrary distinct indices 119894 119895 the generalized lines(119894)(1198941015840) and (119895)(1198951015840) share some common point thus the lines

(119894)(119895) (1198941015840)(1198951015840) should lie on some common plane (Figure 4

provides a quick pictorial proof) and so they should have anonempty intersection But the first of these lines belongsto 119901 whereas the second belongs to 1199011015840 which forces theirintersection to lie on 120598 the common line of the distinct planes119901 1199011015840 Since this happens for all indices 119894 = 119895 we are finished

with part (A)The proof of part (B) is a bit longer For 119899 = 1 the

proposition holds vacuously For 119899 = 2 its truth is immediatewhenever at least one of the generalized lines (1)(11015840) (2)(21015840)is not a line whereas if both of them are lines they eithercoincide (both with the common line of 119901 and 1199011015840) and theresults hold trivially or else they are distinct in which casethe four points (1) (11015840) (2) (21015840) do not lie on the same lineBut then the lines (1)(2) (11015840)(21015840) are also distinct and astheir intersection is nonempty by assumption they definea plane on which lie the points (1) (11015840) (2) (21015840) forcing thelines (1)(11015840) (2)(21015840) to lie on this plane too and thus to havenonempty intersection (a single point since these lines do notcoincide)

For 119899 = 3 and assuming that at least one of the generalizedlines (1)(11015840) (2)(21015840) (3)(31015840) is not a line say (3)(31015840) the resultcomes from the truth of case 119899 = 2 since (1)(11015840) (2)(21015840)have to share some common point which of course lies onthe generalized line (3)(31015840) as well So assume that all threeof these generalized lines are lines (Figure 1) which is theclassical assumption for the converse of DesarguesrsquoTheorem

Observe that at least one of the points (11015840) (21015840) (31015840) say(11015840) does not lie on 119901 otherwise the plane 1199011015840 defined by

these three noncollinear points would coincide with 119901 whichcannot happenThen the line (11015840)(21015840) does not lie on 119901 eitherand so it is distinct from the line (1)(2) which lies on 119901and since (11015840)(21015840) (1)(21015840) have a nonempty intersection theydefine a plane say 119901

12 The line (3)(31015840) cannot lie on 119901

12for

if it did the three noncollinear points (1) (2) (3) would lieon 11990112making it coincide with 119901 and then point (11015840) of 119901

12

would be a point of 119901 a contradiction So the line (3)(31015840) hasa unique common point119874with 119901

12 And we show below that

(3)(31015840) has a unique intersection point with either of lines

4 Geometry

(i)

(i) (i) (i) (i)

(j)

(j)(j) (j) (j)

(i998400) (i998400) (i998400) (i998400)

(i998400)

(j998400)(j998400) (j998400)

(j998400)

(j998400)

Figure 4 Whenever the generalized lines (119894)(1198941015840) (119895)(1198951015840) share a common point the lines (119894)(119895) (1198941015840)(1198951015840) lie on some common plane

(1)(11015840) (2)(2

1015840) of 119901

12 Therefore (1)(11015840) (2)(21015840) (3)(31015840) have

to go through a common point 119874 as wanted Now for theclaim about the intersections of (3)(31015840) with (1)(11015840) (21015840)(21015840)the lines (1)(3) (11015840)(31015840) are distinct otherwise the lines(1)(11015840) (3)(3

1015840) would coincide forcing (3)(31015840) to lie on 119901

12

which it cannot do Then since by hypothesis (1)(3) (11015840)(31015840)intersect nonvacuously they define a plane on which all fourpoints (1) (3) (11015840) (31015840) lie making the lines (1)(11015840) (3)(31015840) lieon it and so have a nonempty intersection This intersectionis actually a unique point otherwise the lines (1)(11015840) (3)(31015840)would coincide forcing (3)(31015840) to lie on 119901

12 a contradiction

Quite similarly (3)(31015840) intersects (21015840)(21015840) in a unique point aswanted

For 119899 ge 4 there exist at least 119899 minus 2 ge 2 lines among the(119894)(1198941015840)rsquos for if not at least three (119894)rsquos would coincide with the

corresponding (1198941015840)rsquos and all these (119894) would be collinear lyingon the common line of 119901 and 1199011015840 contrary to the propositionrsquosassumption

If there exist exactly two lines among the (119894)(1198941015840)rsquos (whichcan happen only for 119899 = 4) say (1)(11015840) and (2)(21015840) thenfor 119894 gt 2 the points (119894) and (1198941015840) coincide and lie on thecommon line 120598 of 119901 1199011015840 Now the lines (1)(2) (11015840)(21015840) cannotcoincide otherwise they would lie on both 119901 1199011015840 and sothey would coincide with 120598 thus all points (119894) would lie on120598 contrary to our original assumption Since (1)(2) (11015840)(21015840)are distinct and by assumption intersect nonvacuously theydefine a plane on which lie all four points (1) (11015840) (2) (21015840)So both lines (1)(11015840) (2)(21015840) lie on this plane as well and theymust have a nonempty intersection a single point Of courseall generalized lines (119894)(1198941015840) go through this point as well

Finally if there exist at least three lines among the (119894)(1198941015840)rsquossay (1)(11015840) (119896)(1198961015840) with 119896 ge 3 then according to theproof of case 119899 = 3 for any 119895 3 le 119895 le 119896 the lines(1)(11015840) (2)(2

1015840) and (119895)(1198951015840) are distinct and intersect each

other in a single point which makes all three of them gothrough a common point say 119874

119895 Since 119874

119895is the unique

common point of the distinct lines (see scholium below)(1)(11015840) (2)(2

1015840) for all indices 119895 all these points coincide and

the lines (1)(11015840) (119896)(1198961015840) go through this common pointwhich of course is also a point of any generalized (119894)(1198941015840) aswanted finishing the proof of the proposition

Remark 2 The above proof reveals that with the classicalhypothesis for the converse of Desarguesrsquo Theorem the threelines (1)(11015840) (2)(21015840) (3)(31015840) are distinct and the three planesdefined by two of them at a time are distinct a result to beused in the next section Indeed with the assumptions madein the proof line (3)(31015840) is distinct from the other two since

(1)

(2)

(3) (4)

p

p998400

(1998400)

(2998400)(4998400)

(3998400)

Figure 5 If (1)(11015840) (2)(21015840) (3)(31015840) are concurrent and (2)(21015840)(3)(31015840) (4)(4

1015840) are concurrent then all (1)(11015840) (2)(21015840) (3)(31015840)

(4)(41015840) are concurrent

it does not lie on the plane 11990112

defined by them whereas if(1)(11015840) (2)(2

1015840) were not distinct the points (1) (11015840) (2) (21015840)

would lie on the same line and this line would be of course(1)(2) forcing (11015840) to lie on (1)(2) and therefore on 119901 acontradiction And any two of the planes defined by two of(1)(11015840) (2)(2

1015840) (3)(3

1015840) have to be distinct since otherwise all

points (1) (2) (3) (11015840) (21015840) (31015840) would lie on the same planeforcing 119901 and 1199011015840 to coincide which cannot be

3 A Generalization for the Converse ofDesarguesrsquo Theorem for Triangle-ConnectedGiven Points

The assumptions postulated in part (B) of Proposition 1 canbe considerably relaxed For example there is no reason forassuming that all lines (119894)(119895) intersect with their correspond-ing lines (1198941015840)(1198951015840) since amuch smaller number of lines amongthe (119894)(119895)rsquos with the same property force all the rest to behavesimilarly

This is made clear in Figure 5 where the lines of the sidesof triangle (1)(2)(3) intersect their corresponding lines oftriangle (11015840)(21015840)(31015840) making the lines (1)(11015840) (2)(21015840) (3)(31015840)go through the same point 119874

1 and similarly the lines of the

sides of triangle (2)(3)(4) intersect their corresponding linesof triangle (21015840)(31015840)(41015840)making the lines (2)(21015840) (3)(31015840) (4)(41015840)go through the same point 119874

2 Since both 119874

1 1198742coin-

cide with the intersection point of (1)(11015840) and (2)(21015840) theyhave to coincide with each other thus making all lines(1)(11015840) (2)(2

1015840) (3)(3

1015840) (4)(4

1015840) go through a common point

Then by part (A) of Proposition 1 lines (1)(4) (11015840)(41015840) haveto intersect as well Of course all the intersections of the pairsof lines (119894)(119895) (1198941015840)(1198951015840) have to lie on the common line of 119901 and1199011015840

Geometry 5

(1)(3)

(2)

(n)

(n minus 1)

(n minus 2)

(a)

(1)

(2)

(3)

(4)

(5)

(n)

(n minus 1)

(n minus 2)

(b)

Figure 6 (a) A triangle-connection set for the points (1) (2) (119899) (b) 119899 minus 4 in number lines can never be all the triangle-connection linesfor 119899 points no three of which are collinear

Also note that the points (1) (2) (3) and (4) need not benoncollinear by three in the argument above we only neededthe existence of the triangles (1)(2)(3) and (2)(3)(4) and theircorresponding ones in 1199011015840 Finally note that if we expand thegiven sets of points with some more points (5) = (51015840) (6) =(61015840) all in the common line of 119901 and 1199011015840 then trivially all

lines (1)(11015840) (2)(21015840) continue to share a common pointThis argument extends similarly for any 119899 ge 3 given

points (119894) and (1198941015840) on the two planes All we need is toassume that all points (119894) that do not coincide with theircorresponding points (1198941015840) are similarly triangle-connected onthe two planes The meaning of this should be rather clear

A triangle-connection sequence in 119878 or for simplicity justtriangle-connection sequence for two points 119860 119861 of a givenset 119878 will be any finite sequence of triangles formed by linesthrough the points of 119878 so that any two consecutive trianglesshare a common side and 119860 belongs to the first triangle ofthis sequence whereas 119861 belongs to the last one recall fromSection 1 that a triangle is a set of three lines A triangle-connection set for 119878 will be a set of triangle-connectionsequences of points of 119878 that contains exactly one suchsequence for each pair of points of 119878 Of coursewhenever sucha set exists for 119878 it is not necessarily unique The triangles inthe sequences of such a set will be called connection trianglesfor 119878 and the lines of the sides connection lines Two givensets of points (1) (119899) and (11015840) (1198991015840) on two planes119901 1199011015840where (119894) corresponds to (1198941015840) will be called similarly triangle-connected whenever there exists a triangle-connection set forthe (119894)rsquos which becomes a triangle-connection set for the (1198941015840)rsquosby replacing all vertices of all triangles in the sequences of thisset by their corresponding points on the other plane

For example a triangle-connection set for the set 119878 =(1) (2) (119899) in Figure 6(a) is 119878

119894119895| 119894 le 119895 + 2 where

119878119894119895= ((119894)(119894+1)(119894+2) (119894+1)(119894+2)(119894+3) (119895minus2)(119895minus1)(119895)) and

triangles are denoted by the three names of their verticesThetriangle-connection lines in this set are (1)(2) (2)(3) (119899minus1)(119899) and (1)(3) (2)(4) (119899 minus 2)(119899) that is 2119899 minus 3 lines in

total It is interesting to note by the way that the minimumnumber of connection lines in a triangle-connection set oflines for 119899 given points no three of which are collinear is 2119899minus3Figure 6(b) hints at the reason a rigorous proof is easy tocome by

Now for two similarly triangle-connected sets of points(119894) and (1198941015840) on two planes 119901 and 1199011015840 and assuming (1)(2)(3)to be a connection triangle on one of the planes weknow by the remark after Proposition 1 that the lines(1)(11015840) (2)(2

1015840) (3)(3

1015840) are distinct and that all three intersect

in a single point 119874 It is quite trivial to show and we do so inthe next paragraph that every line (119894)(1198941015840) intersects all threelines (1)(11015840) (2)(21015840) (3)(31015840) nonvacuously for all 119894 Since for119894 = 1 the line (119894)(1198941015840) does not belong to at least one of thedistinct planes defined by (1)(11015840) (2)(21015840) and (1)(11015840) (3)(31015840)we get that (119894)(1198941015840) has a unique point with at least one ofthe planes In other words (119894)(1198941015840) either intersects (1)(11015840) and(2)(21015840) of the first of these planes on their common point119874 or

else intersects (1)(11015840) and (3)(31015840) of the second plane on theircommon point 119874 So all lines (119894)(1198941015840) go through 119874 provingProposition 3 below which is a generalization of the converseof Desarguesrsquo Theorem

About the claim that for all 119894 the line (119894)(1198941015840) intersectsnonvacuously all three lines (1)(11015840) (2)(21015840) (3)(31015840) for (119894) =(1198941015840) this is immediate so consider an 119894 so that (119894)(1198941015840) is a

usual line and choose a triangle-connection sequence for (1)and (119894) For any triangle (119894

1)(1198942)(1198943) in this sequence and its

corresponding (11989410158401)(1198941015840

2)(1198941015840

3) on the other plane the assumptions

for the converse of the classical Desargues Theorem aresatisfied so (see the scholium) the lines (119894

1)(1198941015840

1) (1198942)(1198941015840

2) (1198943)(1198941015840

3)

are distinct and all three intersect in a single point Now fortwo such consecutive triangles (119894

1)(1198942)(1198943) (1198942)(1198943)(1198944) the two

concurrency points are the same since they both coincidewith the unique intersection point of (119894

2)(1198941015840

2) and (119894

3)(1198941015840

3)This

implies that all lines (119895)(1198951015840) for all the vertices (119895) (1198951015840) of alltriangles in these triangle-connection sequences have to gothrough a common point which implies (1)(11015840) and (119894)(1198941015840)

6 Geometry

p

(1) (4)(3)(2)

p998400

(4998400)(3998400)

(2998400)(1998400)

Figure 7The converse of DesarguesrsquoTheorem does not always holdin case all given points are collinear in both planes

have some common point Similarly (119894)(1198941015840) intersects (2)(21015840)and (3)(31015840) as well and we are done

Proposition 3 Let119901 1199011015840 be two distinct planes (1) (2) (119899)distinct points on 119901 and (11015840) (21015840) (1198991015840) distinct points on 1199011015840where (119894) (1198941015840) are considered to be corresponding Assume thatfor the given points on the two planes which do not coincidewith their corresponding points there exist similar triangle-connection sets so that every pair of corresponding triangle-connection lines (119894)(119895) (1198941015840)(1198951015840) has a nonempty intersectionThen all generalized lines (119894)(1198941015840) go through a common pointMoreover all pairs of lines (119894)(119895) (1198941015840)(1198951015840) for 119894 = 119895 intersectnonvacuously (on the common line of 119901 and 1199011015840)

The existence of the triangle-connection sets guaranteesthat 119899 ge 3 and that at least three points on each of theplanes do not coincide with their corresponding points onthe other If exactly two points on each plane do not coincidewith their corresponding points then all but two lines (119894)(119894)are generalized but the remaining two are not guaranteed tohave a nonempty intersection and thus this case cannot beincluded to the proposition

4 Whenever Three or More Collinear PointsAre Allowed but Corresponding Points AreAssumed Distinct

We close with a third generalization of the converse ofDesarguesrsquoTheorem In Proposition 1 the given points on thetwo planes were allowed to coincide with their correspondingpoints but no three given points on any of the planes wereallowed to be collinear Flipping over these assumptions andrestrictions we now forbid any two corresponding pointsto coincide but allow three or more of them on any of theplanes to be collinear Figure 7 reveals that the converse ofDesarguesrsquo Theorem does not always hold then at least itdoes not hold whenever the given points on the planes areall collinear Interestingly enough it does hold whenever thegiven points on one of the planes do not all lie in a single line

Proposition 4 Let 119901 1199011015840 be two distinct planes (1) (2) (119899) 119899 ge 3 distinct points on 119901 and (11015840) (21015840) (1198991015840) distinctpoints on 1199011015840 so that (119894) = (1198941015840) for all 119894 If all intersections(119894)(119895) cap (119894

1015840)(1198951015840) are nonempty and not all of the given points

are collinear on at least one of the given planes then all lines(119894)(1198941015840) go through a common point

Proof For 119899 = 3 this is exactly part (B) of Proposition 1Similarly it is part (B) of Proposition 1whenever 119899 = 4 and nothree collinear points exist among the given ones in any of theplanes For 119899 = 4 and assuming (1) (2) (3) to be collinear on119901 then (4) does not lie on the line of the other three and so(1)(2)(4) (2)(4)(3) are usual nontrivial triangles the points(1) (2) (3) (4) are triangle-connected and our propositionholds because of Proposition 3

Finally for 119899 ge 5 we proceed inductively assuming theresult holds for all 119896 le 119899 by the propositionrsquos assumptionswe can consider (119894) to be not all collinear If all but oneof them are collinear say (1) not on the line of the restthen (1)(2)(3) (1)(3)(4) (1)(119899 minus 1)(1) form a triangle-connection set for the (119894)rsquos and the result follows fromProposition 3

If on the other hand no more than 119899 minus 2 of the (119894)rsquos arecollinear choose two of them say (1) (2) not on the commonline 120598 of 119901 1199011015840 Now observe that the two sets of 119899 minus 1 innumber points (1) (2) (119899minus2) (119899minus1) and (1) (2) (119899minus2) (119899) satisfy the assumptions of the proposition thus byour induction hypothesis the result holds for them In otherwords all lines (2)(21015840) (3)(31015840) (119899 minus 2)(119899 minus 21015840) (119899 minus 1)(119899 minus11015840) go through a common point 119874

1of (1)(11015840) (2)(21015840) and

similarly all lines (2)(21015840) (3)(31015840) (119899 minus 2)(119899 minus 21015840) (119899)(1198991015840)go through a point 119874

2of (1)(11015840) (2)(21015840) But points 119874

1 1198742

coincide because the lines (1)(11015840) (2)(21015840) are distinct if not(1)(11015840) (2)(2

1015840) would coincide with 120598 because they would be

the line (1)(2) of 119901 and at the same time the line (11015840)(21015840) of1199011015840 this would make (1) (2) points of 120598 a contradiction So all

lines (119894)(1198941015840) go through 1198741and we are done

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] H Dorrie 100 Great Problems of ElementaryMathematicsTheirHistory and Solution Dover 1965

[2] W R Knorr The Ancient Tradition of Geometric ProblemsDover Publications New York NY USA 1993

[3] M Kline Mathematics in Western Culture Oxford UniversityPress 1967

[4] W V Hodge and D PedoeMethods of Algebraic Geometry vol1 Cambridge University Press Cambridge UK 1953

[5] A Pogorelov Geometry Mir Publishers Moscow Russia 1987[6] D Hilbert Grundlagen der Geometrie 1899 (translated asThe

Foundations of Geometry and first published in English byTheOpenCourt Publishing Company Chicago 1992 and offered as anebookbyProject Gutenberg 2005 httpswwwgutenbergorg)

[7] I Bachmann Aufbau der Geometrie aus dem SpiegelungsbegriffGrundlehren derMathematischenWissenschaften 96 SpringerBerlin Germany 2nd edition 1973

[8] H S M Coxeter Introduction to Geometry JohnWiley amp SonsInc New York NY USA 2nd edition 1989

Geometry 7

[9] K Levenberg ldquoA class of non-desarguesian plane geometriesrdquoThe American Mathematical Monthly vol 57 pp 381ndash387 1950

[10] F Moulton ldquoA simple non-Desarguesian plane geometryrdquoTransactions of the American Mathematical Society vol 3 pp102ndash195 1901

[11] K Sitaram ldquoA real non-desarguesian planerdquo The AmericanMathematical Monthly vol 70 pp 522ndash525 1963

[12] OVeblen and JHMaclagan-Wedderburn ldquoNon-Desarguesianand non-Pascalian geometriesrdquo Transactions of the AmericanMathematical Society vol 8 no 3 pp 379ndash388 1907

[13] M Hall ldquoProjective planesrdquo Transactions of the AmericanMathematical Society vol 54 pp 229ndash277 1943

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Geometry

(i)

(i) (i) (i) (i)

(j)

(j)(j) (j) (j)

(i998400) (i998400) (i998400) (i998400)

(i998400)

(j998400)(j998400) (j998400)

(j998400)

(j998400)

Figure 4 Whenever the generalized lines (119894)(1198941015840) (119895)(1198951015840) share a common point the lines (119894)(119895) (1198941015840)(1198951015840) lie on some common plane

(1)(11015840) (2)(2

1015840) of 119901

12 Therefore (1)(11015840) (2)(21015840) (3)(31015840) have

to go through a common point 119874 as wanted Now for theclaim about the intersections of (3)(31015840) with (1)(11015840) (21015840)(21015840)the lines (1)(3) (11015840)(31015840) are distinct otherwise the lines(1)(11015840) (3)(3

1015840) would coincide forcing (3)(31015840) to lie on 119901

12

which it cannot do Then since by hypothesis (1)(3) (11015840)(31015840)intersect nonvacuously they define a plane on which all fourpoints (1) (3) (11015840) (31015840) lie making the lines (1)(11015840) (3)(31015840) lieon it and so have a nonempty intersection This intersectionis actually a unique point otherwise the lines (1)(11015840) (3)(31015840)would coincide forcing (3)(31015840) to lie on 119901

12 a contradiction

Quite similarly (3)(31015840) intersects (21015840)(21015840) in a unique point aswanted

For 119899 ge 4 there exist at least 119899 minus 2 ge 2 lines among the(119894)(1198941015840)rsquos for if not at least three (119894)rsquos would coincide with the

corresponding (1198941015840)rsquos and all these (119894) would be collinear lyingon the common line of 119901 and 1199011015840 contrary to the propositionrsquosassumption

If there exist exactly two lines among the (119894)(1198941015840)rsquos (whichcan happen only for 119899 = 4) say (1)(11015840) and (2)(21015840) thenfor 119894 gt 2 the points (119894) and (1198941015840) coincide and lie on thecommon line 120598 of 119901 1199011015840 Now the lines (1)(2) (11015840)(21015840) cannotcoincide otherwise they would lie on both 119901 1199011015840 and sothey would coincide with 120598 thus all points (119894) would lie on120598 contrary to our original assumption Since (1)(2) (11015840)(21015840)are distinct and by assumption intersect nonvacuously theydefine a plane on which lie all four points (1) (11015840) (2) (21015840)So both lines (1)(11015840) (2)(21015840) lie on this plane as well and theymust have a nonempty intersection a single point Of courseall generalized lines (119894)(1198941015840) go through this point as well

Finally if there exist at least three lines among the (119894)(1198941015840)rsquossay (1)(11015840) (119896)(1198961015840) with 119896 ge 3 then according to theproof of case 119899 = 3 for any 119895 3 le 119895 le 119896 the lines(1)(11015840) (2)(2

1015840) and (119895)(1198951015840) are distinct and intersect each

other in a single point which makes all three of them gothrough a common point say 119874

119895 Since 119874

119895is the unique

common point of the distinct lines (see scholium below)(1)(11015840) (2)(2

1015840) for all indices 119895 all these points coincide and

the lines (1)(11015840) (119896)(1198961015840) go through this common pointwhich of course is also a point of any generalized (119894)(1198941015840) aswanted finishing the proof of the proposition

Remark 2 The above proof reveals that with the classicalhypothesis for the converse of Desarguesrsquo Theorem the threelines (1)(11015840) (2)(21015840) (3)(31015840) are distinct and the three planesdefined by two of them at a time are distinct a result to beused in the next section Indeed with the assumptions madein the proof line (3)(31015840) is distinct from the other two since

(1)

(2)

(3) (4)

p

p998400

(1998400)

(2998400)(4998400)

(3998400)

Figure 5 If (1)(11015840) (2)(21015840) (3)(31015840) are concurrent and (2)(21015840)(3)(31015840) (4)(4

1015840) are concurrent then all (1)(11015840) (2)(21015840) (3)(31015840)

(4)(41015840) are concurrent

it does not lie on the plane 11990112

defined by them whereas if(1)(11015840) (2)(2

1015840) were not distinct the points (1) (11015840) (2) (21015840)

would lie on the same line and this line would be of course(1)(2) forcing (11015840) to lie on (1)(2) and therefore on 119901 acontradiction And any two of the planes defined by two of(1)(11015840) (2)(2

1015840) (3)(3

1015840) have to be distinct since otherwise all

points (1) (2) (3) (11015840) (21015840) (31015840) would lie on the same planeforcing 119901 and 1199011015840 to coincide which cannot be

3 A Generalization for the Converse ofDesarguesrsquo Theorem for Triangle-ConnectedGiven Points

The assumptions postulated in part (B) of Proposition 1 canbe considerably relaxed For example there is no reason forassuming that all lines (119894)(119895) intersect with their correspond-ing lines (1198941015840)(1198951015840) since amuch smaller number of lines amongthe (119894)(119895)rsquos with the same property force all the rest to behavesimilarly

This is made clear in Figure 5 where the lines of the sidesof triangle (1)(2)(3) intersect their corresponding lines oftriangle (11015840)(21015840)(31015840) making the lines (1)(11015840) (2)(21015840) (3)(31015840)go through the same point 119874

1 and similarly the lines of the

sides of triangle (2)(3)(4) intersect their corresponding linesof triangle (21015840)(31015840)(41015840)making the lines (2)(21015840) (3)(31015840) (4)(41015840)go through the same point 119874

2 Since both 119874

1 1198742coin-

cide with the intersection point of (1)(11015840) and (2)(21015840) theyhave to coincide with each other thus making all lines(1)(11015840) (2)(2

1015840) (3)(3

1015840) (4)(4

1015840) go through a common point

Then by part (A) of Proposition 1 lines (1)(4) (11015840)(41015840) haveto intersect as well Of course all the intersections of the pairsof lines (119894)(119895) (1198941015840)(1198951015840) have to lie on the common line of 119901 and1199011015840

Geometry 5

(1)(3)

(2)

(n)

(n minus 1)

(n minus 2)

(a)

(1)

(2)

(3)

(4)

(5)

(n)

(n minus 1)

(n minus 2)

(b)

Figure 6 (a) A triangle-connection set for the points (1) (2) (119899) (b) 119899 minus 4 in number lines can never be all the triangle-connection linesfor 119899 points no three of which are collinear

Also note that the points (1) (2) (3) and (4) need not benoncollinear by three in the argument above we only neededthe existence of the triangles (1)(2)(3) and (2)(3)(4) and theircorresponding ones in 1199011015840 Finally note that if we expand thegiven sets of points with some more points (5) = (51015840) (6) =(61015840) all in the common line of 119901 and 1199011015840 then trivially all

lines (1)(11015840) (2)(21015840) continue to share a common pointThis argument extends similarly for any 119899 ge 3 given

points (119894) and (1198941015840) on the two planes All we need is toassume that all points (119894) that do not coincide with theircorresponding points (1198941015840) are similarly triangle-connected onthe two planes The meaning of this should be rather clear

A triangle-connection sequence in 119878 or for simplicity justtriangle-connection sequence for two points 119860 119861 of a givenset 119878 will be any finite sequence of triangles formed by linesthrough the points of 119878 so that any two consecutive trianglesshare a common side and 119860 belongs to the first triangle ofthis sequence whereas 119861 belongs to the last one recall fromSection 1 that a triangle is a set of three lines A triangle-connection set for 119878 will be a set of triangle-connectionsequences of points of 119878 that contains exactly one suchsequence for each pair of points of 119878 Of coursewhenever sucha set exists for 119878 it is not necessarily unique The triangles inthe sequences of such a set will be called connection trianglesfor 119878 and the lines of the sides connection lines Two givensets of points (1) (119899) and (11015840) (1198991015840) on two planes119901 1199011015840where (119894) corresponds to (1198941015840) will be called similarly triangle-connected whenever there exists a triangle-connection set forthe (119894)rsquos which becomes a triangle-connection set for the (1198941015840)rsquosby replacing all vertices of all triangles in the sequences of thisset by their corresponding points on the other plane

For example a triangle-connection set for the set 119878 =(1) (2) (119899) in Figure 6(a) is 119878

119894119895| 119894 le 119895 + 2 where

119878119894119895= ((119894)(119894+1)(119894+2) (119894+1)(119894+2)(119894+3) (119895minus2)(119895minus1)(119895)) and

triangles are denoted by the three names of their verticesThetriangle-connection lines in this set are (1)(2) (2)(3) (119899minus1)(119899) and (1)(3) (2)(4) (119899 minus 2)(119899) that is 2119899 minus 3 lines in

total It is interesting to note by the way that the minimumnumber of connection lines in a triangle-connection set oflines for 119899 given points no three of which are collinear is 2119899minus3Figure 6(b) hints at the reason a rigorous proof is easy tocome by

Now for two similarly triangle-connected sets of points(119894) and (1198941015840) on two planes 119901 and 1199011015840 and assuming (1)(2)(3)to be a connection triangle on one of the planes weknow by the remark after Proposition 1 that the lines(1)(11015840) (2)(2

1015840) (3)(3

1015840) are distinct and that all three intersect

in a single point 119874 It is quite trivial to show and we do so inthe next paragraph that every line (119894)(1198941015840) intersects all threelines (1)(11015840) (2)(21015840) (3)(31015840) nonvacuously for all 119894 Since for119894 = 1 the line (119894)(1198941015840) does not belong to at least one of thedistinct planes defined by (1)(11015840) (2)(21015840) and (1)(11015840) (3)(31015840)we get that (119894)(1198941015840) has a unique point with at least one ofthe planes In other words (119894)(1198941015840) either intersects (1)(11015840) and(2)(21015840) of the first of these planes on their common point119874 or

else intersects (1)(11015840) and (3)(31015840) of the second plane on theircommon point 119874 So all lines (119894)(1198941015840) go through 119874 provingProposition 3 below which is a generalization of the converseof Desarguesrsquo Theorem

About the claim that for all 119894 the line (119894)(1198941015840) intersectsnonvacuously all three lines (1)(11015840) (2)(21015840) (3)(31015840) for (119894) =(1198941015840) this is immediate so consider an 119894 so that (119894)(1198941015840) is a

usual line and choose a triangle-connection sequence for (1)and (119894) For any triangle (119894

1)(1198942)(1198943) in this sequence and its

corresponding (11989410158401)(1198941015840

2)(1198941015840

3) on the other plane the assumptions

for the converse of the classical Desargues Theorem aresatisfied so (see the scholium) the lines (119894

1)(1198941015840

1) (1198942)(1198941015840

2) (1198943)(1198941015840

3)

are distinct and all three intersect in a single point Now fortwo such consecutive triangles (119894

1)(1198942)(1198943) (1198942)(1198943)(1198944) the two

concurrency points are the same since they both coincidewith the unique intersection point of (119894

2)(1198941015840

2) and (119894

3)(1198941015840

3)This

implies that all lines (119895)(1198951015840) for all the vertices (119895) (1198951015840) of alltriangles in these triangle-connection sequences have to gothrough a common point which implies (1)(11015840) and (119894)(1198941015840)

6 Geometry

p

(1) (4)(3)(2)

p998400

(4998400)(3998400)

(2998400)(1998400)

Figure 7The converse of DesarguesrsquoTheorem does not always holdin case all given points are collinear in both planes

have some common point Similarly (119894)(1198941015840) intersects (2)(21015840)and (3)(31015840) as well and we are done

Proposition 3 Let119901 1199011015840 be two distinct planes (1) (2) (119899)distinct points on 119901 and (11015840) (21015840) (1198991015840) distinct points on 1199011015840where (119894) (1198941015840) are considered to be corresponding Assume thatfor the given points on the two planes which do not coincidewith their corresponding points there exist similar triangle-connection sets so that every pair of corresponding triangle-connection lines (119894)(119895) (1198941015840)(1198951015840) has a nonempty intersectionThen all generalized lines (119894)(1198941015840) go through a common pointMoreover all pairs of lines (119894)(119895) (1198941015840)(1198951015840) for 119894 = 119895 intersectnonvacuously (on the common line of 119901 and 1199011015840)

The existence of the triangle-connection sets guaranteesthat 119899 ge 3 and that at least three points on each of theplanes do not coincide with their corresponding points onthe other If exactly two points on each plane do not coincidewith their corresponding points then all but two lines (119894)(119894)are generalized but the remaining two are not guaranteed tohave a nonempty intersection and thus this case cannot beincluded to the proposition

4 Whenever Three or More Collinear PointsAre Allowed but Corresponding Points AreAssumed Distinct

We close with a third generalization of the converse ofDesarguesrsquoTheorem In Proposition 1 the given points on thetwo planes were allowed to coincide with their correspondingpoints but no three given points on any of the planes wereallowed to be collinear Flipping over these assumptions andrestrictions we now forbid any two corresponding pointsto coincide but allow three or more of them on any of theplanes to be collinear Figure 7 reveals that the converse ofDesarguesrsquo Theorem does not always hold then at least itdoes not hold whenever the given points on the planes areall collinear Interestingly enough it does hold whenever thegiven points on one of the planes do not all lie in a single line

Proposition 4 Let 119901 1199011015840 be two distinct planes (1) (2) (119899) 119899 ge 3 distinct points on 119901 and (11015840) (21015840) (1198991015840) distinctpoints on 1199011015840 so that (119894) = (1198941015840) for all 119894 If all intersections(119894)(119895) cap (119894

1015840)(1198951015840) are nonempty and not all of the given points

are collinear on at least one of the given planes then all lines(119894)(1198941015840) go through a common point

Proof For 119899 = 3 this is exactly part (B) of Proposition 1Similarly it is part (B) of Proposition 1whenever 119899 = 4 and nothree collinear points exist among the given ones in any of theplanes For 119899 = 4 and assuming (1) (2) (3) to be collinear on119901 then (4) does not lie on the line of the other three and so(1)(2)(4) (2)(4)(3) are usual nontrivial triangles the points(1) (2) (3) (4) are triangle-connected and our propositionholds because of Proposition 3

Finally for 119899 ge 5 we proceed inductively assuming theresult holds for all 119896 le 119899 by the propositionrsquos assumptionswe can consider (119894) to be not all collinear If all but oneof them are collinear say (1) not on the line of the restthen (1)(2)(3) (1)(3)(4) (1)(119899 minus 1)(1) form a triangle-connection set for the (119894)rsquos and the result follows fromProposition 3

If on the other hand no more than 119899 minus 2 of the (119894)rsquos arecollinear choose two of them say (1) (2) not on the commonline 120598 of 119901 1199011015840 Now observe that the two sets of 119899 minus 1 innumber points (1) (2) (119899minus2) (119899minus1) and (1) (2) (119899minus2) (119899) satisfy the assumptions of the proposition thus byour induction hypothesis the result holds for them In otherwords all lines (2)(21015840) (3)(31015840) (119899 minus 2)(119899 minus 21015840) (119899 minus 1)(119899 minus11015840) go through a common point 119874

1of (1)(11015840) (2)(21015840) and

similarly all lines (2)(21015840) (3)(31015840) (119899 minus 2)(119899 minus 21015840) (119899)(1198991015840)go through a point 119874

2of (1)(11015840) (2)(21015840) But points 119874

1 1198742

coincide because the lines (1)(11015840) (2)(21015840) are distinct if not(1)(11015840) (2)(2

1015840) would coincide with 120598 because they would be

the line (1)(2) of 119901 and at the same time the line (11015840)(21015840) of1199011015840 this would make (1) (2) points of 120598 a contradiction So all

lines (119894)(1198941015840) go through 1198741and we are done

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] H Dorrie 100 Great Problems of ElementaryMathematicsTheirHistory and Solution Dover 1965

[2] W R Knorr The Ancient Tradition of Geometric ProblemsDover Publications New York NY USA 1993

[3] M Kline Mathematics in Western Culture Oxford UniversityPress 1967

[4] W V Hodge and D PedoeMethods of Algebraic Geometry vol1 Cambridge University Press Cambridge UK 1953

[5] A Pogorelov Geometry Mir Publishers Moscow Russia 1987[6] D Hilbert Grundlagen der Geometrie 1899 (translated asThe

Foundations of Geometry and first published in English byTheOpenCourt Publishing Company Chicago 1992 and offered as anebookbyProject Gutenberg 2005 httpswwwgutenbergorg)

[7] I Bachmann Aufbau der Geometrie aus dem SpiegelungsbegriffGrundlehren derMathematischenWissenschaften 96 SpringerBerlin Germany 2nd edition 1973

[8] H S M Coxeter Introduction to Geometry JohnWiley amp SonsInc New York NY USA 2nd edition 1989

Geometry 7

[9] K Levenberg ldquoA class of non-desarguesian plane geometriesrdquoThe American Mathematical Monthly vol 57 pp 381ndash387 1950

[10] F Moulton ldquoA simple non-Desarguesian plane geometryrdquoTransactions of the American Mathematical Society vol 3 pp102ndash195 1901

[11] K Sitaram ldquoA real non-desarguesian planerdquo The AmericanMathematical Monthly vol 70 pp 522ndash525 1963

[12] OVeblen and JHMaclagan-Wedderburn ldquoNon-Desarguesianand non-Pascalian geometriesrdquo Transactions of the AmericanMathematical Society vol 8 no 3 pp 379ndash388 1907

[13] M Hall ldquoProjective planesrdquo Transactions of the AmericanMathematical Society vol 54 pp 229ndash277 1943

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Geometry 5

(1)(3)

(2)

(n)

(n minus 1)

(n minus 2)

(a)

(1)

(2)

(3)

(4)

(5)

(n)

(n minus 1)

(n minus 2)

(b)

Figure 6 (a) A triangle-connection set for the points (1) (2) (119899) (b) 119899 minus 4 in number lines can never be all the triangle-connection linesfor 119899 points no three of which are collinear

Also note that the points (1) (2) (3) and (4) need not benoncollinear by three in the argument above we only neededthe existence of the triangles (1)(2)(3) and (2)(3)(4) and theircorresponding ones in 1199011015840 Finally note that if we expand thegiven sets of points with some more points (5) = (51015840) (6) =(61015840) all in the common line of 119901 and 1199011015840 then trivially all

lines (1)(11015840) (2)(21015840) continue to share a common pointThis argument extends similarly for any 119899 ge 3 given

points (119894) and (1198941015840) on the two planes All we need is toassume that all points (119894) that do not coincide with theircorresponding points (1198941015840) are similarly triangle-connected onthe two planes The meaning of this should be rather clear

A triangle-connection sequence in 119878 or for simplicity justtriangle-connection sequence for two points 119860 119861 of a givenset 119878 will be any finite sequence of triangles formed by linesthrough the points of 119878 so that any two consecutive trianglesshare a common side and 119860 belongs to the first triangle ofthis sequence whereas 119861 belongs to the last one recall fromSection 1 that a triangle is a set of three lines A triangle-connection set for 119878 will be a set of triangle-connectionsequences of points of 119878 that contains exactly one suchsequence for each pair of points of 119878 Of coursewhenever sucha set exists for 119878 it is not necessarily unique The triangles inthe sequences of such a set will be called connection trianglesfor 119878 and the lines of the sides connection lines Two givensets of points (1) (119899) and (11015840) (1198991015840) on two planes119901 1199011015840where (119894) corresponds to (1198941015840) will be called similarly triangle-connected whenever there exists a triangle-connection set forthe (119894)rsquos which becomes a triangle-connection set for the (1198941015840)rsquosby replacing all vertices of all triangles in the sequences of thisset by their corresponding points on the other plane

For example a triangle-connection set for the set 119878 =(1) (2) (119899) in Figure 6(a) is 119878

119894119895| 119894 le 119895 + 2 where

119878119894119895= ((119894)(119894+1)(119894+2) (119894+1)(119894+2)(119894+3) (119895minus2)(119895minus1)(119895)) and

triangles are denoted by the three names of their verticesThetriangle-connection lines in this set are (1)(2) (2)(3) (119899minus1)(119899) and (1)(3) (2)(4) (119899 minus 2)(119899) that is 2119899 minus 3 lines in

total It is interesting to note by the way that the minimumnumber of connection lines in a triangle-connection set oflines for 119899 given points no three of which are collinear is 2119899minus3Figure 6(b) hints at the reason a rigorous proof is easy tocome by

Now for two similarly triangle-connected sets of points(119894) and (1198941015840) on two planes 119901 and 1199011015840 and assuming (1)(2)(3)to be a connection triangle on one of the planes weknow by the remark after Proposition 1 that the lines(1)(11015840) (2)(2

1015840) (3)(3

1015840) are distinct and that all three intersect

in a single point 119874 It is quite trivial to show and we do so inthe next paragraph that every line (119894)(1198941015840) intersects all threelines (1)(11015840) (2)(21015840) (3)(31015840) nonvacuously for all 119894 Since for119894 = 1 the line (119894)(1198941015840) does not belong to at least one of thedistinct planes defined by (1)(11015840) (2)(21015840) and (1)(11015840) (3)(31015840)we get that (119894)(1198941015840) has a unique point with at least one ofthe planes In other words (119894)(1198941015840) either intersects (1)(11015840) and(2)(21015840) of the first of these planes on their common point119874 or

else intersects (1)(11015840) and (3)(31015840) of the second plane on theircommon point 119874 So all lines (119894)(1198941015840) go through 119874 provingProposition 3 below which is a generalization of the converseof Desarguesrsquo Theorem

About the claim that for all 119894 the line (119894)(1198941015840) intersectsnonvacuously all three lines (1)(11015840) (2)(21015840) (3)(31015840) for (119894) =(1198941015840) this is immediate so consider an 119894 so that (119894)(1198941015840) is a

usual line and choose a triangle-connection sequence for (1)and (119894) For any triangle (119894

1)(1198942)(1198943) in this sequence and its

corresponding (11989410158401)(1198941015840

2)(1198941015840

3) on the other plane the assumptions

for the converse of the classical Desargues Theorem aresatisfied so (see the scholium) the lines (119894

1)(1198941015840

1) (1198942)(1198941015840

2) (1198943)(1198941015840

3)

are distinct and all three intersect in a single point Now fortwo such consecutive triangles (119894

1)(1198942)(1198943) (1198942)(1198943)(1198944) the two

concurrency points are the same since they both coincidewith the unique intersection point of (119894

2)(1198941015840

2) and (119894

3)(1198941015840

3)This

implies that all lines (119895)(1198951015840) for all the vertices (119895) (1198951015840) of alltriangles in these triangle-connection sequences have to gothrough a common point which implies (1)(11015840) and (119894)(1198941015840)

6 Geometry

p

(1) (4)(3)(2)

p998400

(4998400)(3998400)

(2998400)(1998400)

Figure 7The converse of DesarguesrsquoTheorem does not always holdin case all given points are collinear in both planes

have some common point Similarly (119894)(1198941015840) intersects (2)(21015840)and (3)(31015840) as well and we are done

Proposition 3 Let119901 1199011015840 be two distinct planes (1) (2) (119899)distinct points on 119901 and (11015840) (21015840) (1198991015840) distinct points on 1199011015840where (119894) (1198941015840) are considered to be corresponding Assume thatfor the given points on the two planes which do not coincidewith their corresponding points there exist similar triangle-connection sets so that every pair of corresponding triangle-connection lines (119894)(119895) (1198941015840)(1198951015840) has a nonempty intersectionThen all generalized lines (119894)(1198941015840) go through a common pointMoreover all pairs of lines (119894)(119895) (1198941015840)(1198951015840) for 119894 = 119895 intersectnonvacuously (on the common line of 119901 and 1199011015840)

The existence of the triangle-connection sets guaranteesthat 119899 ge 3 and that at least three points on each of theplanes do not coincide with their corresponding points onthe other If exactly two points on each plane do not coincidewith their corresponding points then all but two lines (119894)(119894)are generalized but the remaining two are not guaranteed tohave a nonempty intersection and thus this case cannot beincluded to the proposition

4 Whenever Three or More Collinear PointsAre Allowed but Corresponding Points AreAssumed Distinct

We close with a third generalization of the converse ofDesarguesrsquoTheorem In Proposition 1 the given points on thetwo planes were allowed to coincide with their correspondingpoints but no three given points on any of the planes wereallowed to be collinear Flipping over these assumptions andrestrictions we now forbid any two corresponding pointsto coincide but allow three or more of them on any of theplanes to be collinear Figure 7 reveals that the converse ofDesarguesrsquo Theorem does not always hold then at least itdoes not hold whenever the given points on the planes areall collinear Interestingly enough it does hold whenever thegiven points on one of the planes do not all lie in a single line

Proposition 4 Let 119901 1199011015840 be two distinct planes (1) (2) (119899) 119899 ge 3 distinct points on 119901 and (11015840) (21015840) (1198991015840) distinctpoints on 1199011015840 so that (119894) = (1198941015840) for all 119894 If all intersections(119894)(119895) cap (119894

1015840)(1198951015840) are nonempty and not all of the given points

are collinear on at least one of the given planes then all lines(119894)(1198941015840) go through a common point

Proof For 119899 = 3 this is exactly part (B) of Proposition 1Similarly it is part (B) of Proposition 1whenever 119899 = 4 and nothree collinear points exist among the given ones in any of theplanes For 119899 = 4 and assuming (1) (2) (3) to be collinear on119901 then (4) does not lie on the line of the other three and so(1)(2)(4) (2)(4)(3) are usual nontrivial triangles the points(1) (2) (3) (4) are triangle-connected and our propositionholds because of Proposition 3

Finally for 119899 ge 5 we proceed inductively assuming theresult holds for all 119896 le 119899 by the propositionrsquos assumptionswe can consider (119894) to be not all collinear If all but oneof them are collinear say (1) not on the line of the restthen (1)(2)(3) (1)(3)(4) (1)(119899 minus 1)(1) form a triangle-connection set for the (119894)rsquos and the result follows fromProposition 3

If on the other hand no more than 119899 minus 2 of the (119894)rsquos arecollinear choose two of them say (1) (2) not on the commonline 120598 of 119901 1199011015840 Now observe that the two sets of 119899 minus 1 innumber points (1) (2) (119899minus2) (119899minus1) and (1) (2) (119899minus2) (119899) satisfy the assumptions of the proposition thus byour induction hypothesis the result holds for them In otherwords all lines (2)(21015840) (3)(31015840) (119899 minus 2)(119899 minus 21015840) (119899 minus 1)(119899 minus11015840) go through a common point 119874

1of (1)(11015840) (2)(21015840) and

similarly all lines (2)(21015840) (3)(31015840) (119899 minus 2)(119899 minus 21015840) (119899)(1198991015840)go through a point 119874

2of (1)(11015840) (2)(21015840) But points 119874

1 1198742

coincide because the lines (1)(11015840) (2)(21015840) are distinct if not(1)(11015840) (2)(2

1015840) would coincide with 120598 because they would be

the line (1)(2) of 119901 and at the same time the line (11015840)(21015840) of1199011015840 this would make (1) (2) points of 120598 a contradiction So all

lines (119894)(1198941015840) go through 1198741and we are done

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] H Dorrie 100 Great Problems of ElementaryMathematicsTheirHistory and Solution Dover 1965

[2] W R Knorr The Ancient Tradition of Geometric ProblemsDover Publications New York NY USA 1993

[3] M Kline Mathematics in Western Culture Oxford UniversityPress 1967

[4] W V Hodge and D PedoeMethods of Algebraic Geometry vol1 Cambridge University Press Cambridge UK 1953

[5] A Pogorelov Geometry Mir Publishers Moscow Russia 1987[6] D Hilbert Grundlagen der Geometrie 1899 (translated asThe

Foundations of Geometry and first published in English byTheOpenCourt Publishing Company Chicago 1992 and offered as anebookbyProject Gutenberg 2005 httpswwwgutenbergorg)

[7] I Bachmann Aufbau der Geometrie aus dem SpiegelungsbegriffGrundlehren derMathematischenWissenschaften 96 SpringerBerlin Germany 2nd edition 1973

[8] H S M Coxeter Introduction to Geometry JohnWiley amp SonsInc New York NY USA 2nd edition 1989

Geometry 7

[9] K Levenberg ldquoA class of non-desarguesian plane geometriesrdquoThe American Mathematical Monthly vol 57 pp 381ndash387 1950

[10] F Moulton ldquoA simple non-Desarguesian plane geometryrdquoTransactions of the American Mathematical Society vol 3 pp102ndash195 1901

[11] K Sitaram ldquoA real non-desarguesian planerdquo The AmericanMathematical Monthly vol 70 pp 522ndash525 1963

[12] OVeblen and JHMaclagan-Wedderburn ldquoNon-Desarguesianand non-Pascalian geometriesrdquo Transactions of the AmericanMathematical Society vol 8 no 3 pp 379ndash388 1907

[13] M Hall ldquoProjective planesrdquo Transactions of the AmericanMathematical Society vol 54 pp 229ndash277 1943

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Geometry

p

(1) (4)(3)(2)

p998400

(4998400)(3998400)

(2998400)(1998400)

Figure 7The converse of DesarguesrsquoTheorem does not always holdin case all given points are collinear in both planes

have some common point Similarly (119894)(1198941015840) intersects (2)(21015840)and (3)(31015840) as well and we are done

Proposition 3 Let119901 1199011015840 be two distinct planes (1) (2) (119899)distinct points on 119901 and (11015840) (21015840) (1198991015840) distinct points on 1199011015840where (119894) (1198941015840) are considered to be corresponding Assume thatfor the given points on the two planes which do not coincidewith their corresponding points there exist similar triangle-connection sets so that every pair of corresponding triangle-connection lines (119894)(119895) (1198941015840)(1198951015840) has a nonempty intersectionThen all generalized lines (119894)(1198941015840) go through a common pointMoreover all pairs of lines (119894)(119895) (1198941015840)(1198951015840) for 119894 = 119895 intersectnonvacuously (on the common line of 119901 and 1199011015840)

The existence of the triangle-connection sets guaranteesthat 119899 ge 3 and that at least three points on each of theplanes do not coincide with their corresponding points onthe other If exactly two points on each plane do not coincidewith their corresponding points then all but two lines (119894)(119894)are generalized but the remaining two are not guaranteed tohave a nonempty intersection and thus this case cannot beincluded to the proposition

4 Whenever Three or More Collinear PointsAre Allowed but Corresponding Points AreAssumed Distinct

We close with a third generalization of the converse ofDesarguesrsquoTheorem In Proposition 1 the given points on thetwo planes were allowed to coincide with their correspondingpoints but no three given points on any of the planes wereallowed to be collinear Flipping over these assumptions andrestrictions we now forbid any two corresponding pointsto coincide but allow three or more of them on any of theplanes to be collinear Figure 7 reveals that the converse ofDesarguesrsquo Theorem does not always hold then at least itdoes not hold whenever the given points on the planes areall collinear Interestingly enough it does hold whenever thegiven points on one of the planes do not all lie in a single line

Proposition 4 Let 119901 1199011015840 be two distinct planes (1) (2) (119899) 119899 ge 3 distinct points on 119901 and (11015840) (21015840) (1198991015840) distinctpoints on 1199011015840 so that (119894) = (1198941015840) for all 119894 If all intersections(119894)(119895) cap (119894

1015840)(1198951015840) are nonempty and not all of the given points

are collinear on at least one of the given planes then all lines(119894)(1198941015840) go through a common point

Proof For 119899 = 3 this is exactly part (B) of Proposition 1Similarly it is part (B) of Proposition 1whenever 119899 = 4 and nothree collinear points exist among the given ones in any of theplanes For 119899 = 4 and assuming (1) (2) (3) to be collinear on119901 then (4) does not lie on the line of the other three and so(1)(2)(4) (2)(4)(3) are usual nontrivial triangles the points(1) (2) (3) (4) are triangle-connected and our propositionholds because of Proposition 3

Finally for 119899 ge 5 we proceed inductively assuming theresult holds for all 119896 le 119899 by the propositionrsquos assumptionswe can consider (119894) to be not all collinear If all but oneof them are collinear say (1) not on the line of the restthen (1)(2)(3) (1)(3)(4) (1)(119899 minus 1)(1) form a triangle-connection set for the (119894)rsquos and the result follows fromProposition 3

If on the other hand no more than 119899 minus 2 of the (119894)rsquos arecollinear choose two of them say (1) (2) not on the commonline 120598 of 119901 1199011015840 Now observe that the two sets of 119899 minus 1 innumber points (1) (2) (119899minus2) (119899minus1) and (1) (2) (119899minus2) (119899) satisfy the assumptions of the proposition thus byour induction hypothesis the result holds for them In otherwords all lines (2)(21015840) (3)(31015840) (119899 minus 2)(119899 minus 21015840) (119899 minus 1)(119899 minus11015840) go through a common point 119874

1of (1)(11015840) (2)(21015840) and

similarly all lines (2)(21015840) (3)(31015840) (119899 minus 2)(119899 minus 21015840) (119899)(1198991015840)go through a point 119874

2of (1)(11015840) (2)(21015840) But points 119874

1 1198742

coincide because the lines (1)(11015840) (2)(21015840) are distinct if not(1)(11015840) (2)(2

1015840) would coincide with 120598 because they would be

the line (1)(2) of 119901 and at the same time the line (11015840)(21015840) of1199011015840 this would make (1) (2) points of 120598 a contradiction So all

lines (119894)(1198941015840) go through 1198741and we are done

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] H Dorrie 100 Great Problems of ElementaryMathematicsTheirHistory and Solution Dover 1965

[2] W R Knorr The Ancient Tradition of Geometric ProblemsDover Publications New York NY USA 1993

[3] M Kline Mathematics in Western Culture Oxford UniversityPress 1967

[4] W V Hodge and D PedoeMethods of Algebraic Geometry vol1 Cambridge University Press Cambridge UK 1953

[5] A Pogorelov Geometry Mir Publishers Moscow Russia 1987[6] D Hilbert Grundlagen der Geometrie 1899 (translated asThe

Foundations of Geometry and first published in English byTheOpenCourt Publishing Company Chicago 1992 and offered as anebookbyProject Gutenberg 2005 httpswwwgutenbergorg)

[7] I Bachmann Aufbau der Geometrie aus dem SpiegelungsbegriffGrundlehren derMathematischenWissenschaften 96 SpringerBerlin Germany 2nd edition 1973

[8] H S M Coxeter Introduction to Geometry JohnWiley amp SonsInc New York NY USA 2nd edition 1989

Geometry 7

[9] K Levenberg ldquoA class of non-desarguesian plane geometriesrdquoThe American Mathematical Monthly vol 57 pp 381ndash387 1950

[10] F Moulton ldquoA simple non-Desarguesian plane geometryrdquoTransactions of the American Mathematical Society vol 3 pp102ndash195 1901

[11] K Sitaram ldquoA real non-desarguesian planerdquo The AmericanMathematical Monthly vol 70 pp 522ndash525 1963

[12] OVeblen and JHMaclagan-Wedderburn ldquoNon-Desarguesianand non-Pascalian geometriesrdquo Transactions of the AmericanMathematical Society vol 8 no 3 pp 379ndash388 1907

[13] M Hall ldquoProjective planesrdquo Transactions of the AmericanMathematical Society vol 54 pp 229ndash277 1943

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Geometry 7

[9] K Levenberg ldquoA class of non-desarguesian plane geometriesrdquoThe American Mathematical Monthly vol 57 pp 381ndash387 1950

[10] F Moulton ldquoA simple non-Desarguesian plane geometryrdquoTransactions of the American Mathematical Society vol 3 pp102ndash195 1901

[11] K Sitaram ldquoA real non-desarguesian planerdquo The AmericanMathematical Monthly vol 70 pp 522ndash525 1963

[12] OVeblen and JHMaclagan-Wedderburn ldquoNon-Desarguesianand non-Pascalian geometriesrdquo Transactions of the AmericanMathematical Society vol 8 no 3 pp 379ndash388 1907

[13] M Hall ldquoProjective planesrdquo Transactions of the AmericanMathematical Society vol 54 pp 229ndash277 1943

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of