KöMaL 2007 B4009

The angle bisectors drawn from the vertices $A$ and $B$ of a triangle ABC intersect the opposite sides at $A_1$ and $B_1$, respectively. $P$ is the intersection of the ray $A_1 B_1$ with the circumscribed circle of the triangle. Prove that $\frac{1}{PA} = \frac{1}{PB} + \frac{1}{PC}.$