HINTS ON NAPOLEON'S TRIANGLE THEOREM
Construct an equilateral triangle and see if Napoleon's triangle is equilateral. You will see from your construction, that when given an equilateral triangle, the resulting Napoleon triangle is also equilateral. Now, construct an isosceles triangle. Again, you will see that with an isosceles triangle, Napoleon's triangle is still equilateral. Now, construct a right triangle. Still, even with a right triangle, Napoelon's triangle is equilateral.
You may also need to use the Law of Cosines and the below diagram:
Now you can better prove that for any given triangle ABC, Napoleon's triangle is equilateral.