Conic Sections · Challenge problem

An elliptical pool table has foci $F_1 = (-6, 0)$ and $F_2 = (6, 0)$ and semi-major axis $10$. A ball is struck from $F_1$, bounces once off the rail, and (by the reflection property of an ellipse) travels directly to $F_2$. Find the perimeter of the triangle formed by $F_1$, $F_2$, and the point where the ball bounces off the rail.