Mohr-Mascheroni theorem

Transcription

1 Mohr-Mascheroni theorem NOGNENG Dorian LIX October 25, 2016

2 Table of Contents Introduction Constructible values Projection Intersecting a circle with a line Ratio a b c Intersecting 2 lines Conclusion

3 Introduction - Target I show the following : any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone

4 Introduction - Target I show the following : any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone a.k.a. : Mohr-Mascheroni theorem

5 Introduction - Problem statement Some points given on a sheet of paper

6 Introduction - Problem statement Some points given on a sheet of paper We draw lines and circles

7 Introduction - Problem statement We want to do the same, using only a compass (draws circles)

8 Introduction - Problem statement We want to do the same, using only a compass (draws circles) The same means any point: of course, no line.

9 Introduction - Problem statement We want to do the same, using only a compass (draws circles) The same means any point: of course, no line. We assume that we can report lengths

10 Introduction - Plan of action Plan of action: Show that we can construct many lengths: 1, 2, 3, 2, 1 2, etc

11 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 a line

12 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 a line Show that we can construct the ratio of lengths a b c

13 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 a line Show that we can construct the ratio of lengths a b c Show that we can construct the intersection between any circle and any line

14 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 a line Show that we can construct the ratio of lengths a b c Show that we can construct the intersection between any circle and any line Show that we can construct the intersection between any 2 lines

15 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 a line Show that we can construct the ratio of lengths a b c Show that we can construct the intersection between any circle and any line Show that we can construct the intersection between any 2 lines The above steps are enough

16 Table of Contents Introduction Constructible values Projection Intersecting a circle with a line Ratio a b c Intersecting 2 lines Conclusion

17 Constructible values - Any integer n

18 Constructible values - a 2 b 2 for any a, b a and b are any previously known distances.

19 Constructible values - a 2 + b 2 for any a, b If c is any large distance, we can create the following distances: c 2 a 2

20 Constructible values - a 2 + b 2 for any a, b If c is any large distance, we can create the following distances: c 2 a 2 c 2 a 2 b 2 = ( c 2 a 2) 2 b 2

21 Constructible values - a 2 + b 2 for any a, b If c is any large distance, we can create the following distances: c 2 a 2 c 2 a 2 b 2 = ( c 2 a 2) 2 b 2 a 2 + b 2 = c 2 ( c 2 a 2 b 2) 2

22 Constructible values - a + b, a b for any a, b We can align distances.

23 Constructible values - b2 a 2 c a, b, c: known distances.

24 Constructible values Using the above: 1 2 =

25 Table of Contents Introduction Constructible values Projection Intersecting a circle with a line Ratio a b c Intersecting 2 lines Conclusion

26 Projection - Middle of a segment

27 Projection - Middle of a segment We can draw the circle whose diameter is a given segment

28 Projection of a point on a line

29 Table of Contents Introduction Constructible values Projection Intersecting a circle with a line Ratio a b c Intersecting 2 lines Conclusion

30 Intersecting a circle with a line P: projection of A on (BC)

31 Intersecting a circle with a line P: projection of A on (BC) Draw circle centered at P with radius R 2 PA 2.

32 Table of Contents Introduction Constructible values Projection Intersecting a circle with a line Ratio a b c Intersecting 2 lines Conclusion

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

34 Table of Contents Introduction Constructible values Projection Intersecting a circle with a line Ratio a b c Intersecting 2 lines Conclusion

35 Intersecting 2 lines - Reduce to projected points

36 Intersecting 2 lines Notice that EI = EF CE CE, CI = CD CE DF CE DF (Thales)

37 Table of Contents Introduction Constructible values Projection Intersecting a circle with a line Ratio a b c Intersecting 2 lines Conclusion

38 Conclusion We have proven that any point that can be drawn using a compass and straightedge can also be drawn using only a compass The proof can be extended if we do not assume that we can report lengths

39 Conclusion QUESTIONS?