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Napoleon's theorem
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 lines connecting 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 the areas of the outer and inner Napoleon triangles equals the area of the original triangle.
The theorem is often attributed to Napoleon Bonaparte (1769–1821). According to Howard Eves, the theorem and a construction problem bearing Napoleon's name were discovered by his friend and adviser Lorenzo Mascheroni (1750–1800), who let the Emperor claim them for himself. Some have suggested that it may 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.
In the figure above, △ABC is the original triangle. △AZB, △BXC, △CYA are equilateral triangles constructed on its sides' exteriors, and points L, M, 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 △LMN is equilateral is to observe that MN becomes CZ under a clockwise rotation of 30° around A and a homothety of ratio with the same center, and that LN also becomes CZ after a counterclockwise rotation of 30° around B and a homothety of ratio with the same center. The respective spiral similarities are That implies MN = LN and the angle between them must be 60°.
There are in fact many proofs of the theorem's statement, including a synthetic (coordinate-free) one, a trigonometric one, a symmetry-based approach, and proofs using complex numbers.
The theorem has frequently been attributed to Napoleon, but several papers have been written concerning this issue which cast doubt upon this assertion (see (Grünbaum 2012)).
The following entry appeared on page 47 in the Ladies' Diary of 1825 (so in late 1824, a year or so after the compilation of Dublin examination papers). This is an early appearance of Napoleon's theorem in print, and Napoleon's name is not mentioned.
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Napoleon's theorem
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 lines connecting 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 the areas of the outer and inner Napoleon triangles equals the area of the original triangle.
The theorem is often attributed to Napoleon Bonaparte (1769–1821). According to Howard Eves, the theorem and a construction problem bearing Napoleon's name were discovered by his friend and adviser Lorenzo Mascheroni (1750–1800), who let the Emperor claim them for himself. Some have suggested that it may 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.
In the figure above, △ABC is the original triangle. △AZB, △BXC, △CYA are equilateral triangles constructed on its sides' exteriors, and points L, M, 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 △LMN is equilateral is to observe that MN becomes CZ under a clockwise rotation of 30° around A and a homothety of ratio with the same center, and that LN also becomes CZ after a counterclockwise rotation of 30° around B and a homothety of ratio with the same center. The respective spiral similarities are That implies MN = LN and the angle between them must be 60°.
There are in fact many proofs of the theorem's statement, including a synthetic (coordinate-free) one, a trigonometric one, a symmetry-based approach, and proofs using complex numbers.
The theorem has frequently been attributed to Napoleon, but several papers have been written concerning this issue which cast doubt upon this assertion (see (Grünbaum 2012)).
The following entry appeared on page 47 in the Ladies' Diary of 1825 (so in late 1824, a year or so after the compilation of Dublin examination papers). This is an early appearance of Napoleon's theorem in print, and Napoleon's name is not mentioned.
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