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Triangles

(2005-07-26)   Napoleon's Theorem
The equilateral triangles supported by a triangle have equidistant centers.

This theorem is usually intended for equilateral triangles built outside of the base triangle, but it also holds if the three triangles are built inward.

Napoléon's theorem is one of the most rediscovered results of elementary euclidean geometry.  The French ruler  Napoléon Bonaparte  (1769-1821)  certainly had the mathematical ability to discover this for himself, but there's no evidence that he did so.  The theorem first appeared in print in 1825, in an article written for  The Ladies' Diary  by Dr. W. Rutherford.  It may well have been Rutherford himself who decided to name this theorem after the recently deceased French emperor  Napoléon I.

One easy way to prove this is to observe that properly rotating the figure by angles of  ± p/3  (successively) about the centers of two of the equilateral triangles brings the center of the third back to its original position.  This establishes the equality of two sides of the triangle formed by the centers of the 3 equilateral triangles.  Since the same argument holds with any particular choice among such centers, the aforementioned triangle is necessarily  equilateral.  

Napoleon Tiling :

Napoleon's theorem can be made visually obvious with a periodic tiling of the plane like the one which serves as the background for this page.  The black triangles are congruent scalene triangles in three orientations.  The  3  families of equilateral triangles are represented with  3  different colors.

Mathpages  |  Cut-the-Knot  |  MathWorld  |  Wikipedia
GeoGebra Institute of Hong Kong  (Napoleon Tiling)

(2005-07-26)   Morley's Trisector Theorem   (Morley's Miracle, 1899)
The adjacent trisectors of the 3 inner angles meet at 3 equidistant points.

This nice theorem was discovered in 1899 by  Frank Morley (1860-1937).

Mathpages  |  Cut-the-Knot  |  MathWorld  |  Wikipedia  |  Proof by  Alain Connes

(2020-06-16)   Six-point Conway circle of a triangle
Tribute to the recently-deceased  John Conway (1937-2020).  RIP.

For each vertex of a given triangle,  we obtain two new points by extending both sides outward by a distance equal to the side opposite to that vertex.  All six new points are on a circle centered on the triangle's incenter  I  (i.e.,  the point where the three inner angle bisectors intersect).  That circle is called  Conway's circle.

Proof :   Any pair of new points form the base of an isoceles triangle whose apex is a vertex of the original triangle.  Both legs are equal either to an original side of to the sum of two such sides.  Their perpendicular bisector  (the locus of the centers of the circles to which they both belong)  is thus an inner angular bisector of the original triangle.  

Conway circle   |   Deux minutes pour John Conway by  El Jj  (2020-06-11).