ISL 2002 G4

Circles $S_1$ and $S_2$ intersect at points $P$ and $Q$ . Distinct points $A_1$ and $B_1$ (not at $P$ or $Q$ ) are selected on $S_1$ . The lines $A_1P$ and $B_1P$ meet $S_2$ again at $A_2$ and $B_2$ respectively, and the lines $A_1B_1$ and $A_2B_2$ meet at $C$ . Prove that, as $A_1$ and $B_1$ vary, the circumcentres of triangles $A_1A_2C$ all lie on one fixed circle.

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