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IMO ShortList 1999, geometry problem 8

Source: IMO ShortList 1999, geometry problem 8

November 13, 2004
geometrycircumcircleincenterangle bisectorTriangleIMO Shortlist

Problem Statement

Given a triangle ABCABC. The points AA, BB, CC divide the circumcircle Ω\Omega of the triangle ABCABC into three arcs BCBC, CACA, ABAB. Let XX be a variable point on the arc ABAB, and let O1O_{1} and O2O_{2} be the incenters of the triangles CAXCAX and CBXCBX. Prove that the circumcircle of the triangle XO1O2XO_{1}O_{2} intersects the circle Ω\Omega in a fixed point.
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