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Triangles

(2005-07-26)  
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 theoremis one of the most rediscoveredresults 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 therecently deceased French emperor Napoléon I.

One easy way to prove this is to observe that properly rotating the figure by anglesof  /3  (successively)about the centers of two of the equilateral trianglesbrings the center of the third back to its original position. This establishes the equality of two sides of the triangleformed by the centers of the 3 equilateral triangles. Since the same argument holds withany 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 planelike the one which serves as the background for this page. The black triangles are congruent scalene triangles in three orientations. The  3  families of equilateraltriangles are represented with  3  different colors.

(2005-07-26)  
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).

(2020-06-16)  
Tribute to the recently-deceased John Conway (1937-2020).  RIP.

For each vertex of a given triangle,  we obtain two new points by extendingboth 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'sincenter  I (i.e.,  the point where the three inner angle bisectors intersect). That circle is called Conway's circle.

 Any pair of new points form the base of an isoceles triangle whose apexis a vertex of the original triangle.  Both legs are equaleither 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.