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.From an arbitrary point in an equilateraltriangle, segments to the vertices and perpendiculars to the sides partition thetriangle into six smaller triangles A, B, C, D, E, F (see Figure 2.27 left).Claim [183]:A + C + E = B + D + F. Mathematical Properties 49Figure 2.27: Six Triangles [183]Drawing three additional lines through the selected point which are parallelto the sides of the original triangle partitions it into three parallelograms andthree small equilateral triangles (Figure 2.27 right).Since the areas of theparallelograms are bisected by their diagonals and the equilateral triangles bytheir altitudes,A + C + E = x + a + y + b + z + c = B + D + F.Figure 2.28: Pompeiu s Theorem [184]Property 36 (Pompeiu s Theorem).If P is an arbitrary point in an equi-lateral triangle ABC then there exists a triangle with sides of length P A, P B,P C [184].Draw segments P L, P M, P N parallel to the sides of the triangle (Figure2.28).Then, the trapezoids P MAN, P NBL, P LCM are isosceles and thushave equal diagonals.Hence, P A = MN, P B=LN, P C = LM and "LMNis the required triangle.Note that the theorem remains valid for any point Pin the plane of "ABC [184] and that the triangle is degenerate if and only ifP lies on the circumcircle of "ABC [267]. 50 Mathematical PropertiesFigure 2.29: Random Point [177]Property 37 (Random Point).A point P is chosen at random inside anequilateral triangle.Perpendiculars from P to the sides of the triangle meetthese sides at points X, Y , Z.The probability that a triangle with sides P X,1P Y , P Z exists is equal to [177].4As shown in Figure 2.29, the segments satisfy the triangle inequality if andonly if the point lies in the shaded region whose area is one fourth that of theoriginal triangle [122].Compare this result to Pompeiu s Theorem!Property 38 (Gauss Plane).In the Gauss (complex) plane [81], "ABC isequilateral if and only if"(b - a)»2 = (c - b)»± = a - c; »± := (-1 ± 1 3)/2.±For »±, "ABC is described counterclockwise/clockwise, respectively.Property 39 (Gauss Theorem on Triangular Numbers).In his diaryof July 10, 1796, Gauss wrote [290]: E¥P HKA! num = " + " + ". I.e., Eureka! Every positive integer is the sum of at most three triangularnumbers. As early as 1638, Fermat conjectured much more in his polygonal numbertheorem [319]: Every positive integer is a sum of at most three triangularnumbers, four square numbers, five pentagonal numbers, and n n-polygonalnumbers. (Alas, his margin was once again too narrow to hold his proof!)Jacobi and Lagrange proved the square case in 1772, Gauss the triangular casein 1796, and Cauchy the general case in 1813 [320]. Mathematical Properties 51Figure 2.30: Equilateral Shadows [180]Property 40 (Equilateral Shadows).Any triangle can be orthogonally pro-jected onto an equilateral triangle [180].Moreover, under the inverse of thistransformation, the incircle of the equilateral triangle is mapped to the mid-point ellipse of the original triangle with center at the triangle centroid andtangent to the triangle sides at their midpoints (Figure 2.30).Note that this demonstrates that if we cut a triangle from a piece of paperand hold it under the noonday sun then we can always position the triangleso that its shadow is an equilateral triangle.Figure 2.31: Fundamental Theorem of Affine Geometry [33]Property 41 (Affine Geometry).All triangles are affine-congruent [33].Inparticular, any triangle may be affinely mapped onto any equilateral triangle(Figure 2.31).This theorem is of fundamental importance in the theory of Riemann sur-faces.E.g.[288, p.113]:Theorem 2.1 (Riemann Surfaces).If an arbitrary manifold M is givenwhich is both triangulable and orientable then it is possible to define an analyticstructure on M which makes it into a Riemann surface. 52 Mathematical PropertiesFigure 2.32: Largest Inscribed Triangle and Least-Diameter Decomposition ofthe Open Disk [4]Property 42 (Largest Inscribed Triangle).The triangle of largest areathat is inscribed in a given circle is the equilateral triangle (Figure 2.32) [249].Property 43 (Planar Soap Bubble Clusters).An inscribed equilateraltriangle (Figure 2
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