REGULAR POLYGONS. Răzvan Gelca Texas Tech University

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1 REGULAR POLYGONS Răzvan Gelca Texas Tech University

2 Definition. A regular polygon is a polygon in which all sides are equal and all angles are equal.

3 Definition. A regular polygon is a polygon in which all sides are equal and all angles are equal.

4 Definition. A regular polygon is a polygon in which all sides are equal and all angles are equal.

5 Definition. A regular polygon is a polygon in which all sides are equal and all angles are equal.

6 Definition. A regular polygon is a polygon in which all sides are equal and all angles are equal.

7 Definition. A regular polygon is a polygon in which all sides are equal and all angles are equal.

8 Definition. A regular polygon is a polygon in which all sides are equal and all angles are equal.

9 Construct an equilateral triangle.

10 Construct an equilateral triangle.

11 Construct an equilateral triangle.

12 Construct an equilateral triangle.

13 Construct an equilateral triangle.

14 Construct a square.

15 Construct a square.

16 Construct a square.

17 Construct a square.

18 Construct a square.

19 Construct a square.

20 Construct a regular pentagon.

21 Theorem. (Gauss-Wantzel) A regular polygon with n sides can be constructed if an only if the odd prime factors of n are distinct Fermat primes. This means that n = 2 m (2 2k1 + 1)(2 2k2 + 1) (2 2kt + 1), where each 2 2ki + 1 is prime and the k i s are distinct. Examples: regular pentagon 5 = regular heptadecagon 17 = regular polygon with 2570 sides

22 Problem 0. What regular polygons tesselate the plane?

23 Problem 0. What regular polygons tesselate the plane?

24 Problem 0. What regular polygons tesselate the plane?

25 Problem 1. What regular polygons tesselate the plane?

26 Are there others?

27 Are there others? The angles that meet at a point should add up to 360.

28 Are there others? The angles that meet at a point should add up to 360. The angles of a regular n-gon are equal to n 2 n 180.

29 Are there others? The angles that meet at a point should add up to 360. The angles of a regular n-gon are equal to n 2 n 180. Hence n 2 n multiplied by some integer should equal 2. The equality (n 2)k = 2n can only hold for n = 3, k = 6; n = 4, k = 4; n = 6, k = 3.

30 Problem 2. Let ABC be an equilateral triangle and P a point in its interior such that P A = 3, P B = 4, P C = 5. Find the side-length of the triangle.

31 We will use the following result: Pompeiu s Theorem. Let ABC be an equilateral triangle and P a point in its plane. Then there is a triangle whose sides are P A, P B, P C.

32 Here is the proof:

33 Here is the proof:

34 Here is the proof:

35 Here is the proof:

36 Here is the proof:

37 Let us return to the original problem:

38 Let us return to the original problem:

39 Let us return to the original problem:

40 Let us return to the original problem:

41 The Law of Cosines gives AB 2 = cos 150

42 The Law of Cosines gives AB 2 = cos 150 = , and hence AB =

43 Problem 3. Let ABCD be a square and M a point inside it such that MAB = MBA = 15. Find the angle DMC.

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48 Problem 4. Let ABCDE be a regular pentagon and M a point in its interior with the property that MBA = MEA = 42. Find CMD.

49 Let us return to the previous problem. Assume that somehow we guessed that CMD = 60. How can we prove it?

50 Construct instead M such that the triangle DM C is equilateral. Then DA = DM and CB = CM. So the triangles DAM and CBM are isosceles. It follows that DAM = DMA = 75, so M is the point from the statement of the problem.

51 Now let us return to the problem with the regular pentagon. Construct instead the point M such that the triangle CMD is equilateral. Then triangle DEM is isosceles, and EDM = = 48.

52 Thus We get DEM = 1 2 ( ) = 66. AEM = = 42. Similarly MBA = 42 and thus M is the point from the statement of the problem.

53 Problem 5. Nineteen darts hit a target which is a regular hexagon of side-length 1. Show that two of the darts are at distance at most 3/3 from each other.

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57 Problem 6. Prove that Let A 1 A 2 A 3 A 4 A 5 A 6 A 7 be a regular heptagon. 1 A 1 A 2 = 1 A 1 A A 1 A 4.

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60 A 1 OA 2 = 360 7, A 1OA 3 = 720 7, A 1OA 4 =

61 A 1 A 2 = 2R sin 180 7, A 1A 3 = 2R sin 360 7, A 1A 4 = 2R sin

62 So we have to prove that 1 = Rewrite as sin sin 7 7 sin = sin sin sin sin sin sin

63 Now we use the formula 2 sin a sin b = cos(a b) cos(a + b).

64 So we have to prove that 1 = Rewrite as sin sin to write this as sin = sin sin sin sin sin sin cos cos = cos cos cos cos

65 We are left with showing that cos cos 720 cos 900 cos 360 = Note that = 1260 and cos(180 x) = cos x. Hence the left-hand side is zero, as desired.

66 There is a more elegant way to write this, which makes the solution more natural.

67 There is a more elegant way to write this, which makes the solution more natural. Use 180 = π.

68 1 sin π 7 = 1 sin 2π sin 3π 7 sin 2π 7 sin 3π 7 = sin π 7 sin 2π 7 + sin π 7 sin 3π 7 cos 5π 7 + cos π 7 = cos π 7 cos 3π 7 + cos 2π 7 cos 4π 7 cos 3π 7 + cos 4π 7 cos 5π 7 cos 2π 7 = 0 This is the same as cos 3π 7 + cos ( π 3π 7 ) cos 5π7 cos ( π 5π 7 ) = 0 Now use cos(π x) = cos x to conclude that this is true.

69 Problem 7. A regular octagon of side-length 1 is dissected into parallelograms. Find the sum of the areas of the rectangles in the dissection.

70 Problem 7. A regular octagon of side-length 1 is dissected into parallelograms. Find the sum of the areas of the rectangles in the dissection.

71 Problem 7. A regular octagon of side-length 1 is dissected into parallelograms. Find the sum of the areas of the rectangles in the dissection. Answer: 2.

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77 Problem 8. Let A 1 A 2 A 3... A 12 be a regular dodecagon. Prove that A 1 A 5, A 4 A 8, and A 3 A 6 intersect at one point.

78 First let us recall the construction of a regular dodecagon.

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96 Problem 9. On a circle of diameter AB choose points C, D, E on one side of AB and F on the other side such that AC= CD= BE= 20 and BF = 60. Prove that F M = F E.

97 Problem 9. On a circle of diameter AB choose points C, D, E on one side of AB and F on the other side such that AC= CD= BE= 20 and BF = 60. Prove that F M = F E. Where is the regular polygon in this problem???

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109 Problem 10. For what n does there exists an n-gon in the plane all of whose vertices have integer coordinates?

110 Problem 10. For what n does there exists an n-gon in the plane all of whose vertices have integer coordinates?

111 Problem 10. For what n does there exists an n-gon in the plane all of whose vertices have integer coordinates? Are there other regular polygons besides the square?

112 For n > 6, start with the smallest such polygon...

113 and produce a smaller one.

114 If there is a regular n-gon with vertices of integer coordinates, the center has rational coordinates. By changing the scale we can assume that the center has integer coordinates as well. Several 90 rotations around the center produce a regular polygon with 12 or 20 sides with vertices of integer coordinates, which we know cannot exits.

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