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Geometry from Iran TST

Source: Iranian TST 2018, third exam day 2, problem 6

April 19, 2018
geometryIranIranian TST

Problem Statement

Consider quadrilateral ABCDABCD inscribed in circle ω\omega . P≡AC∩BDP\equiv AC\cap BD. EE, FF lie on sides ABAB, CDCD respectively such that APE^=DPF^\hat {APE}=\hat {DPF} . Circles ω1\omega_1, ω2\omega_2 are tangent to ω\omega at XX , YY respectively and also both tangent to the circumcircle of △PEF\triangle PEF at PP . Prove that: EXEY=FXFY\frac {EX}{EY}=\frac {FX}{FY}
Proposed by Ali Zamani
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