Question 71
The angle $\angle BAC$ of triangle $ABC$ is $30^\circ$, and $P$ is an arbitrary point in the interior of this triangle. Denote the reflections of the point $P$ in the sides $BC$, $CA$ and $AB$ by $P_A$, $P_B$ and $P_C$, respectively. Suppose that $P_A P_B P_C$ is an equilateral triangle. Prove that then $\angle BPC = 90^\circ$.