Question 10
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_{1}P $ and $ B_{1}P $ meet $ S_{2} $ again at $ A_{2} $ and $ B_{2} $ respectively, and the lines $ A_{1}B_{1} $ and $ A_{2}B_{2} $ meet at C. Prove that, as $ A_{1} $ and $ B_{1} $ vary, the circumcentres of triangles $ A_{1}A_{2}C $ all lie on one fixed circle.