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Pappus’ Theorem for a Conicand Mystic Hexagons
Ross MooreMacquarie University
Sydney, Australia
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Pappus’ Theorem for a Conicand Mystic Hexagons
Ross MooreMacquarie University
Sydney, Australia
Pappus’ Theorem is a well-known result for triples of points on
two lines in the(projective) plane:
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2
Theorem. (Pappus)
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2
Theorem. (Pappus) Given lines l and m in the plane,
l
ggggggggggggggggggggggggggggggggggggggggggggggggggggggg
m
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2
Theorem. (Pappus) Given lines l and m in the plane,three
distinct points A, B, C on l (but not on m),
l⊗
A ⊗B
⊗C ggggggggggggggggggggggggggggggggggggggggggggggggggggggg
m
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2
Theorem. (Pappus) Given lines l and m in the plane,three
distinct points A, B, C on l (but not on m),and three distinct
points D, E, F on m (but not on l),
l⊗
A ⊗B
⊗C ggggggggggggggggggggggggggggggggggggggggggggggggggggggg
m ⊗D ⊗
E⊗
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2
Theorem. (Pappus) Given lines l and m in the plane,three
distinct points A, B, C on l (but not on m),and three distinct
points D, E, F on m (but not on l),
construct the intersections L = BF ∧ CE,
l⊗
A ⊗B
⊗C ggggggggggggggggggggggggggggggggggggggggggggggggggggggg
m ⊗D ⊗
E⊗
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BBBB
BBBB
BBBB
BBBB
BBBB
BBB
~~~~
~~~~
~~~~
~~~~
~~~~
~~~~
~
⊕L
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2
Theorem. (Pappus) Given lines l and m in the plane,three
distinct points A, B, C on l (but not on m),and three distinct
points D, E, F on m (but not on l),
construct the intersections L = BF ∧ CE, M = CD ∧ AF
l⊗
A ⊗B
⊗C ggggggggggggggggggggggggggggggggggggggggggggggggggggggg
m ⊗D ⊗
E⊗
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MMM
⊕M
BBBB
BBBB
BBBB
BBBB
BBBB
BBB
~~~~
~~~~
~~~~
~~~~
~~~~
~~~~
~
⊕L
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2
Theorem. (Pappus) Given lines l and m in the plane,three
distinct points A, B, C on l (but not on m),and three distinct
points D, E, F on m (but not on l),
construct the intersections L = BF ∧ CE, M = CD ∧ AFand N = AE ∧
BD.
l⊗
A ⊗B
⊗C ggggggggggggggggggggggggggggggggggggggggggggggggggggggg
m ⊗D ⊗
E⊗
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::::
::::
::::
::
⊕N
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rrrrrr
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MMM
⊕M
BBBB
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BBBB
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BBB
~~~~
~~~~
~~~~
~~~~
~~~~
~~~~
~
⊕L
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2
Theorem. (Pappus) Given lines l and m in the plane,three
distinct points A, B, C on l (but not on m),and three distinct
points D, E, F on m (but not on l),
construct the intersections L = BF ∧ CE, M = CD ∧ AFand N = AE ∧
BD. Then L, M , N are collinear.
l⊗
A ⊗B
⊗C ggggggggggggggggggggggggggggggggggggggggggggggggggggggg
m ⊗D ⊗
E⊗
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::::
::::
::::
::
⊕N
rrrrrr
rrrrrr
rrrrrr
rrrrrr
rrrrrr
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MMMMMM
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MMMMMM
MMM
⊕M
BBBB
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BBBB
BBBB
BBBB
BBB
~~~~
~~~~
~~~~
~~~~
~~~~
~~~~
~
⊕L \\\\\\\\\\\\\\\\\\\\\\\\\
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3
Definition. By a Pappus configuration we mean a set of 6 points
in theplane, arranged as a pair of triples of points (not
necessarily collinear) (A, B, C)and (D, E, F ), such that the
intersections (L, M, N), of pairs of lines join-ing points taken as
in Pappus’ Theorem, are collinear.
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3
Definition. By a Pappus configuration we mean a set of 6 points
in theplane, arranged as a pair of triples of points (not
necessarily collinear) (A, B, C)and (D, E, F ), such that the
intersections (L, M, N), of pairs of lines join-ing points taken as
in Pappus’ Theorem, are collinear.
Not so well-known is that if the six points A, B, C, D, E, F lie
on anon-singular conic, then (A, B, C), (D, E, F ) form a Pappus
configuration.
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3
Definition. By a Pappus configuration we mean a set of 6 points
in theplane, arranged as a pair of triples of points (not
necessarily collinear) (A, B, C)and (D, E, F ), such that the
intersections (L, M, N), of pairs of lines join-ing points taken as
in Pappus’ Theorem, are collinear.
Not so well-known is that if the six points A, B, C, D, E, F lie
on anon-singular conic, then (A, B, C), (D, E, F ) form a Pappus
configuration.
⊗A ⊗
B⊗
C
⊗
D⊗
E
⊗F
⊕N ⊕
M ⊕L
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4
Remark. The usual Pappus’ Theorem is just the situation whereby
the conicdegenerates into a pair of lines.
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4
Remark. The usual Pappus’ Theorem is just the situation whereby
the conicdegenerates into a pair of lines.
Remark. When the six points are ordered as A, F , B, D, C, F the
result-ing polygon is just Pascal’s “mystic hexagon”.
Alternatively, given a mystichexagon, the Pappus configuration is
obtained by taking the even and oddvertices for the groups of
three.
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4
Remark. The usual Pappus’ Theorem is just the situation whereby
the conicdegenerates into a pair of lines.
Remark. When the six points are ordered as A, F , B, D, C, F the
result-ing polygon is just Pascal’s “mystic hexagon”.
Alternatively, given a mystichexagon, the Pappus configuration is
obtained by taking the even and oddvertices for the groups of
three.
Remark. It is well known that a unique conic can be drawn
through any 5points in general position.
Pappus configurations give a way to construct that conic,
parametrised by thepoints on a line...
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5
Given five points A, B, C, D, E in the plane in sufficiently
general position,
⊗A ⊗
B⊗
C
⊗
D⊗
E
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5
Given five points A, B, C, D, E in the plane in sufficiently
general position,let N = AE ∧ BD.
⊗A ⊗
B⊗
C
⊗
D⊗
E
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5
Given five points A, B, C, D, E in the plane in sufficiently
general position,let N = AE ∧ BD. Pick P along CD.
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B⊗
C
⊗
D⊗
E
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5
Given five points A, B, C, D, E in the plane in sufficiently
general position,let N = AE ∧ BD. Pick P along CD. Let Q = NP ∧
CE.
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5
Given five points A, B, C, D, E in the plane in sufficiently
general position,let N = AE ∧ BD. Pick P along CD. Let Q = NP ∧
CE.Then X = AP ∧ BQ lies on the conic.
⊗A ⊗
B⊗
C
⊗
D⊗
E
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5
Given five points A, B, C, D, E in the plane in sufficiently
general position,let N = AE ∧ BD. Pick P along CD. Let Q = NP ∧
CE.Then X = AP ∧ BQ lies on the conic.Vary P on CD for different
points X on the conic:
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B⊗
C
⊗
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5
Given five points A, B, C, D, E in the plane in sufficiently
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CE.Then X = AP ∧ BQ lies on the conic.Vary P on CD for different
points X on the conic:
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B⊗
C
⊗
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E
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-
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Given five points A, B, C, D, E in the plane in sufficiently
general position,let N = AE ∧ BD. Pick P along CD. Let Q = NP ∧
CE.Then X = AP ∧ BQ lies on the conic.Vary P on CD for different
points X on the conic:
⊗A ⊗
B⊗
C
⊗
D⊗
E
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5
Given five points A, B, C, D, E in the plane in sufficiently
general position,let N = AE ∧ BD. Pick P along CD. Let Q = NP ∧
CE.Then X = AP ∧ BQ lies on the conic.Vary P on CD for different
points X on the conic:
⊗A ⊗
B⊗
C
⊗
D⊗
E
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-
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Given five points A, B, C, D, E in the plane in sufficiently
general position,let N = AE ∧ BD. Pick P along CD. Let Q = NP ∧
CE.Then X = AP ∧ BQ lies on the conic.Vary P on CD for different
points X on the conic:
⊗A ⊗
B⊗
C
⊗
D⊗
E
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-
5
Given five points A, B, C, D, E in the plane in sufficiently
general position,let N = AE ∧ BD. Pick P along CD. Let Q = NP ∧
CE.Then X = AP ∧ BQ lies on the conic.Vary P on CD for different
points X on the conic:
⊗A ⊗
B⊗
C
⊗
D⊗
E
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