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Parallel and not parallel lines.

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April 30, 2017
geometry

Problem Statement

Let ABCDABCD be a convex quadrilateral with ABAB is not parallel to CDCD Circle Γ1\Gamma_{1} with center O1O_1 passes through AA and BB, and touches segment CDCD at PP. Circle Γ2\Gamma_{2} with center O2O_2 passes through CC and DD, and touches segment ABAB at QQ. Let EE and FF be the intersection of circles Γ1\Gamma_{1} and Γ2\Gamma_{2}. Prove that EFEF bisects segment PQPQ if and only if BCBC is parallel to ADAD.
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