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Quadrilateral geo with a condition

Source: MEMO 2023 T5

August 25, 2023
geometry

Problem Statement

We are given a convex quadrilateral ABCDABCD whose angles are not right. Assume there are points P,Q,R,SP, Q, R, S on its sides AB,BC,CD,DAAB, BC, CD, DA, respectively, such that PS∥BDPS \parallel BD, SQ⊥BCSQ \perp BC, PR⊥CDPR \perp CD. Furthermore, assume that the lines PR,SQPR, SQ, and ACAC are concurrent. Prove thatthe points P,Q,R,SP, Q, R, S are concyclic.
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