OMCC 2003 Problem 4

$S_1$ and $S_2$ are two circles that intersect at two different points $P$ and $Q$ . Let $\ell_1$ and $\ell_2$ be two parallel lines such that $\ell_1$ passes through the point $P$ and intersects $S_1,S_2$ at $A_1,A_2$ respectively (both distinct from $P$ ), and $\ell_2$ passes through the point $Q$ and intersects $S_1,S_2$ at $B_1,B_2$ respectively (both distinct from $Q$ ). Show that the triangles $A_1QA_2$ and $B_1PB_2$ have the same perimeter.