CR-MO 2003 Problem 4

$S_{1}$ and $S_{2}$ are two circles that intersect at distinct points $P$ and $Q$ . $\ell_{1}$ and $\ell_{2}$ are two parallel lines through $P$ and $Q$ . $\ell_{1}$ intersects $S_{1}$ and $S_{2}$ at points $A_{1}$ and $A_{2}$ , different from $P$ , respectively. $\ell_{2}$ intersects $S_{1}$ and $S_{2}$ at points $B_{1}$ and $B_{2}$ , different from $Q$ , respectively. Show that the perimeters of the triangles $A_{1}QA_{2}$ and $B_{1}PB_{2}$ are equal.