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In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the centres of those equilateral triangles themselves form an equilateral triangle. The triangle thus formed is called the inner or outer ''Napoleon triangle''. The difference in area of these two triangles equals the area of the original triangle. The theorem is often attributed to Napoleon Bonaparte (1769–1821). Some have suggested that it may just date back to W. Rutherford's 1825 question published in ''The Ladies' Diary'', four years after the French emperor's death, but the result is covered in three questions set in an examination for a Gold Medal at the University of Dublin in October, 1820, whereas Napoleon died the following May. ==Proofs== In the figure above, ABC is the original triangle. AZB, BXC, and CYA are equilateral triangles constructed on its sides' exteriors, and points L, M, and N are the centroids of those triangles. The theorem for outer triangles states that triangle LMN ''(green)'' is equilateral. A quick way to see that the triangle LMN is equilateral is to observe that MN becomes CZ under a clockwise rotation of 30° around A and a homothety of ratio √''3'' with the same center, and that LN also becomes CZ after a counterclockwise rotation of 30° around B and a homothety of ratio √''3'' with the same center. The respective spiral similarities are A(√''3'',-30°) and B(√''3'',30°). That implies MN = LN and the angle between them must be 60°.〔For a visual demonstration see ''(Napoleon's Theorem via Two Rotations )'' at Cut-the-Knot.〕 There are in fact many proofs of the theorem's statement, including a trigonometric one,〔 a symmetry-based approach, and proofs using complex numbers.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Napoleon's theorem」の詳細全文を読む スポンサード リンク
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