Videos:
The Triangle Book has yet to appear
by John Conway
(MOVES 2015).
Of triangles, gases, prices and men
by Cedric Villani (2015年05月05日).
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. QED
Napoleon Bonaparte (1769-1821)
Napoleon Bonaparte (1643-1727)
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)
This nice theorem was discovered in 1899 by Frank Morley (1860-1937).
[画像: Come back later, we're still working on this one... ]
Mathpages | Cut-the-Knot | MathWorld | Wikipedia | Proof by Alain Connes
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. QED
Conway circle | Deux minutes pour John Conway by El Jj (2020年06月11日).
