A Geometry Challenge

I recently had some fun with one of my favorite triangles.  It all started with this innocent NY State Regents Exam question:

In triangle ABC, we have a = 15, b = 14, and c = 13.  Find the measure of angle C.

This problem is designed to test the student’s knowledge of the Law of Cosines, but because of the special nature of the 13-14-15 triangle, it’s easy to find angle C without it.

And since it was a multiple choice question, I considered another approach.  I constructed an equilateral triangle with side AB and produced the following diagram:

Before I actually performed the construction, I assumed that third vertex of the equilateral triangle would lie in the interior of the original triangle ABC.  By the construction above, it appears to be outside triangle ABC.

So here’s the challenge:  prove that the third vertex of the equilateral triangle lies inside triangle ABC without using the Law of Cosines!