Given any triangle ABC, construct equilateral triangles on each side and find the center of each equilateral triangle. The triangle formed by these three centers is

It appears that Napoleon's triangle is equilateral. Can we prove it?

Consider the following construction.

With the use of some trigonometry, we can prove that Napoleon's triangle is equilateral...

Let's begin with what we know.

Now, let's apply the Law of Cosines.

Substituting our values of x and y above, we get:

Expanding the cosine of the sum gives:

Substituting this expansion into our previous equation yields:

Next, apply the Law of Cosines to the original triangle ABC.

Remember that we also know:

(from the Law of Sines)

Substituting this into our equation results in:

Because our final equation is symmetrical in a, b, and c, we can conclude that Napoleon's triangle is equilateral!

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