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Geometric inequality

Source: Turkish TST 2011 Problem 7

July 23, 2011
inequalitiesgeometrycircumcircleratiogeometric transformationhomothetyinequalities proposed

Problem Statement

Let KK be a point in the interior of an acute triangle ABCABC and ARBPCQARBPCQ be a convex hexagon whose vertices lie on the circumcircle Γ\Gamma of the triangle ABC.ABC. Let A1A_1 be the second point where the circle passing through KK and tangent to Γ\Gamma at AA intersects the line AP.AP. The points B1B_1 and C1C_1 are defined similarly. Prove that min⁡{PA1AA1, QB1BB1, RC1CC1}≤1. \min\left\{\frac{PA_1}{AA_1}, \: \frac{QB_1}{BB_1}, \: \frac{RC_1}{CC_1}\right\} \leq 1.
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