ISL 1999 Problem 8

Given a triangle $ABC$ . The points $A$ , $B$ , $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$ , $CA$ , $AB$ . Let $X$ be a variable point on the arc $AB$ , and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$ . Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.

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