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Folding quadrilaterals

Source: 2022 Israel National Olympiad P5

December 16, 2022
geometry3D geometry

Problem Statement

A paper convex quadrilateral will be called folding if there are points P,Q,R,SP,Q,R,S on the interiors of segments AB,BC,CD,DAAB,BC,CD,DA respectively so that if we fold in the triangles SAP,PBQ,QCR,RDSSAP, PBQ, QCR, RDS, they will exactly cover the quadrilateral PQRSPQRS. In other words, if the folded triangles will cover the quadrilateral PQRSPQRS but won't cover each other. Prove that if quadrilateral ABCDABCD is folding, then AC⊥BDAC\perp BD or ABCDABCD is a trapezoid.
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