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Prove circumcircle tangent to segments

Source: APMO 1999

March 18, 2006
geometrycircumcirclegeometric transformation

Problem Statement

Let Γ1\Gamma_1 and Γ2\Gamma_2 be two circles intersecting at PP and QQ. The common tangent, closer to PP, of Γ1\Gamma_1 and Γ2\Gamma_2 touches Γ1\Gamma_1 at AA and Γ2\Gamma_2 at BB. The tangent of Γ1\Gamma_1 at PP meets Γ2\Gamma_2 at CC, which is different from PP, and the extension of APAP meets BCBC at RR. Prove that the circumcircle of triangle PQRPQR is tangent to BPBP and BRBR.
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