IL-GILLIS 2022 Problem 5

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

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