BALTIC-WAY 1990 Problem 8

Let $P$ be a point on the circumcircle of a triangle $A B C$. It is known that the base points of the perpendiculars drawn from $P$ onto the lines $A B, B C$ and $C A$ lie on one straight line (called a Simson line). Prove that the Simson lines of two diametrically opposite points $P_{1}$ and $P_{2}$ are perpendicular.