In mathematics, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centroids of those equilateral triangles themselves form an equilateral triangle.
The triangle thus formed is called the Napoleon triangle (inner and outer). 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). However, it may just date back to W. Rutherford's 1825 publication The Ladies' Diary, four years after the French emperor's death.[1]
A quick way to see that the triangle LMN is equilateral is to observe that MN and LN have the same image (namely CZ) under the respective spiral similarities A(√3,-30°) and B(√3,30°). That implies MN = LN and the angle between them must be 60°.
See also
External links
- Napoleon's Theorem at MathPages
- Napoleon's Theorem and Generalizations
- To see the construction
- Napoleon's Theorem by Jay Warendorff, The Wolfram Demonstrations Project.
- Eric W. Weisstein, Napoleon's Theorem at MathWorld.
- Eric W. Weisstein, Spiral Similarity at MathWorld.
- Napoleon's Theorem and some generalizations, variations & converses at Dynamic Geometry Sketches
This article incorporates material from Napoleon's theorem on PlanetMath, which is licensed under the GFDL.
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