Napoleon’s Theorem

I came across the Napoleon’s Theorem when I found my CS Professer’s Puzzle Corner. Here is the problem description from Wikipedia, 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.

Proof

Verification

To finally show that the distances are the same, I was not brave enough to simplify that expression. Instead, I plugged it into an online simplifer. Here is link1 and link2 which shows that all the distances are indeed the same expression.

Solution Page

You can check out another solution to this problem on the May-June issue of Puzzle Corner. My solution was acknowledged in the Other Responders section.