Mohr-Mascheroni theorem

NOGNENG Dorian

LIX

October 25, 2016

Table of Contents

Introduction

Constructible values

Projection

Intersecting a circle with a line

Ratio a·bc

Intersecting 2 lines

Conclusion

Introduction - Target

I show the following :

• any geometric construction that can be performed by acompass and straightedge can be performed by a compassalone

• a.k.a. : Mohr-Mascheroni theorem

Introduction - Target

I show the following :

• any geometric construction that can be performed by acompass and straightedge can be performed by a compassalone

• a.k.a. : Mohr-Mascheroni theorem

Introduction - Problem statement

• Some points given on a sheet of paper

• We draw lines and circles

Introduction - Problem statement

• Some points given on a sheet of paper

• We draw lines and circles

Introduction - Problem statement

• We want to do the same, using only a compass (drawscircles)

• ”The same” means any point: of course, no line.

• We assume that we can report lengths

Introduction - Problem statement

• We want to do the same, using only a compass (drawscircles)

• ”The same” means any point: of course, no line.

• We assume that we can report lengths

Introduction - Problem statement

• We want to do the same, using only a compass (drawscircles)

• ”The same” means any point: of course, no line.

• We assume that we can report lengths

Introduction - Plan of action

Plan of action:• Show that we can construct many lengths:1, 2, 3,

√2, 1

2, etc

• Show that we can construct the projection of a point on aline

• Show that we can construct the ratio of lengths a·bc

• Show that we can construct the intersection between anycircle and any line

• Show that we can construct the intersection between any 2lines

• The above steps are enough

Introduction - Plan of action

Plan of action:• Show that we can construct many lengths:1, 2, 3,

√2, 1

2, etc

• Show that we can construct the projection of a point on aline

• Show that we can construct the ratio of lengths a·bc

• Show that we can construct the intersection between anycircle and any line

• Show that we can construct the intersection between any 2lines

• The above steps are enough

Introduction - Plan of action

Plan of action:• Show that we can construct many lengths:1, 2, 3,

√2, 1

2, etc

• Show that we can construct the projection of a point on aline

• Show that we can construct the ratio of lengths a·bc

• Show that we can construct the intersection between anycircle and any line

• Show that we can construct the intersection between any 2lines

• The above steps are enough

Introduction - Plan of action

Plan of action:• Show that we can construct many lengths:1, 2, 3,

√2, 1

2, etc

• Show that we can construct the projection of a point on aline

• Show that we can construct the ratio of lengths a·bc

• Show that we can construct the intersection between anycircle and any line

• Show that we can construct the intersection between any 2lines

• The above steps are enough

Introduction - Plan of action

Plan of action:• Show that we can construct many lengths:1, 2, 3,

√2, 1

2, etc

• Show that we can construct the projection of a point on aline

• Show that we can construct the ratio of lengths a·bc

• Show that we can construct the intersection between anycircle and any line

• Show that we can construct the intersection between any 2lines

• The above steps are enough

Introduction - Plan of action

Plan of action:• Show that we can construct many lengths:1, 2, 3,

√2, 1

2, etc

• Show that we can construct the projection of a point on aline

• Show that we can construct the ratio of lengths a·bc

• Show that we can construct the intersection between anycircle and any line

• Show that we can construct the intersection between any 2lines

• The above steps are enough

Table of Contents

Introduction

Constructible values

Projection

Intersecting a circle with a line

Ratio a·bc

Intersecting 2 lines

Conclusion

Constructible values - Any integer n

Constructible values -√a2 − b2 for any a, b

a and b are any previously known distances.

Constructible values -√a2 + b2 for any a, b

If c is any large distance, we can create the following distances:

•√c2 − a2

•√c2 − a2 − b2 =

√(√c2 − a2

)2 − b2

•√a2 + b2 =

√c2 −

(√c2 − a2 − b2

)2

Constructible values -√a2 + b2 for any a, b

If c is any large distance, we can create the following distances:

•√c2 − a2

•√c2 − a2 − b2 =

√(√c2 − a2

)2 − b2

•√a2 + b2 =

√c2 −

(√c2 − a2 − b2

)2

Constructible values -√a2 + b2 for any a, b

If c is any large distance, we can create the following distances:

•√c2 − a2

•√c2 − a2 − b2 =

√(√c2 − a2

)2 − b2

•√a2 + b2 =

√c2 −

(√c2 − a2 − b2

)2

Constructible values - a + b, a − b for any a, b

We can align distances.

Constructible values - b2−a2c

a, b, c : known distances.

Constructible values - 12

Using the above:1

2=

22 − 12

2− 1

Table of Contents

Introduction

Constructible values

Projection

Intersecting a circle with a line

Ratio a·bc

Intersecting 2 lines

Conclusion

Projection - Middle of a segment

⇒ We can draw the circle whose diameter is a given segment

Projection - Middle of a segment

⇒ We can draw the circle whose diameter is a given segment

Projection of a point on a line

Table of Contents

Introduction

Constructible values

Projection

Intersecting a circle with a line

Ratio a·bc

Intersecting 2 lines

Conclusion

Intersecting a circle with a line

P : projection of A on (BC )

Draw circle centered at P with radius√R2 − PA2.

Intersecting a circle with a line

P : projection of A on (BC )

Draw circle centered at P with radius√

R2 − PA2.

Table of Contents

Introduction

Constructible values

Projection

Intersecting a circle with a line

Ratio a·bc

Intersecting 2 lines

Conclusion

Ratio a·bc

Draw circle of diameter of length b and find G at distance cfrom A.Then project C on (AG ) (C at distance a from A).

Table of Contents

Introduction

Constructible values

Projection

Intersecting a circle with a line

Ratio a·bc

Intersecting 2 lines

Conclusion

Intersecting 2 lines - Reduce to projected points

Intersecting 2 lines

Notice that EI = EF · CECE−DF

, CI = CD · CECE−DF

(Thales)

Table of Contents

Introduction

Constructible values

Projection

Intersecting a circle with a line

Ratio a·bc

Intersecting 2 lines

Conclusion

Conclusion

• We have proven that any point that can be drawn using acompass and straightedge can also be drawn using only acompass• The proof can be extended if we do not assume that we canreport lengths

Conclusion

QUESTIONS ?