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A, Q, R are colinear

Source: Own. IMO 2021 Malaysian Training Camp 1

December 31, 2020
geometry

Problem Statement

Let ABCABC be an actue triangle with AB<ACAB<AC. Let Γ\Gamma be its circumcircle, II its incenter and PP is a point on Γ\Gamma such that ∠API=90∘\angle API=90^{\circ}. Let QQ be a point on Γ\Gamma such that QB⋅tan⁡∠B=QC⋅tan⁡∠CQB\cdot\tan \angle B=QC\cdot \tan \angle C Consider a point RR such that PRPR is tangent to Γ\Gamma and BR=CRBR=CR. Prove that the points A,Q,RA, Q, R are colinear.
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