Three Circles in a Triangle: A Problem

This week, I was playing around with some geometry and came up with a fun problem.

Suppose there is a triangle, $\Delta ABC$. Inside the triangle are three circles, each of which is tangent to the other circles and to two sides of the triangle (see diagram below).

We can connect the centers of the three circles to make a new triangle, $\Delta PQR$.

Problem: Show that if $\Delta PQR$ is similar to $\Delta ABC$, then $\Delta ABC$ is equilateral.