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Standard triangle geometry: BQ bisects CA

Source: problem 9 (G3) of QEDMO 1; created by myself

November 7, 2005
geometrycircumcirclepower of a pointradical axisgeometry proposed

Problem Statement

Let ABCABC be a triangle with AB≠CBAB\neq CB. Let C′C^{\prime} be a point on the ray [AB[AB such that AC′=CBAC^{\prime}=CB. Let A′A^{\prime} be a point on the ray [CB[CB such that CA′=ABCA^{\prime}=AB. Let the circumcircles of triangles ABA′ABA^{\prime} and CBC′CBC^{\prime} intersect at a point QQ (apart from BB). Prove that the line BQBQ bisects the segment CACA. Darij
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