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Parallelism in a cyclic quad, circle & diagonals diagram

Source: XVIII Tuymaada Mathematical Olympiad (2011), Junior Level

July 29, 2011

geometrycircumcirclecyclic quadrilateralprojective geometrygeometry unsolved

Problem Statement

A circle passing through the vertices

A

and

B

of a cyclic quadrilateral

ABCD

intersects diagonals

AC

and

BD

E

and

F

, respectively. The lines

AF

and

BC

meet at a point

P

, and the lines

BE

and

AD

meet at a point

Q

. Prove that

PQ

is parallel to

CD