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it is bisect iff the other is parallel

Source: unknown

September 24, 2014
geometryparallelogrampower of a pointradical axis

Problem Statement

Let ABCDABCD be a convex quadrilateral with ABAB is not parallel to CDCD. Circle ω1\omega_1 with center O1O_1 passes through AA and BB, and touches segment CDCD at PP. Circle ω2\omega_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 ω1\omega_1 and ω2\omega_2. Prove that EFEF bisects segment PQPQ if and only if BCBC is parallel to ADAD.
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