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BQCR is # iff AC=BC

Source: 2024 Mathematics Regional Olympiad of Mexico West P4

October 21, 2024
geometryisoscelesparallelogram

Problem Statement

Let △ABC\triangle ABC be a triangle and ω\omega its circumcircle. The tangent to ω\omega through BB cuts the parallel to BCBC through AA at PP. The line CPCP cuts the circumcircle of △ABP\triangle ABP again in QQ and line AQAQ cuts ω\omega at RR. Prove that BQCRBQCR is parallelogram if and only if AC=BCAC=BC.
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