Proofs
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 an homothety of ratio √3 with the same center, and that LN also becomes CZ after a counterclockwise rotation of 30° around B and an 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°.
Analytically, it can be determined that each of the three segments of the LMN triangle has a length of:
There are in fact many proofs of the theorem's statement, including a trigonometric one, a symmetry-based approach, and proofs using complex numbers.
Read more about this topic: Napoleon's Theorem
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—David Hume (1711–1776)
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—William Shakespeare (1564–1616)