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tangent circles wanted, starting with tangential ABCD

Source: 2021 Sharygin Geometry Olympiad Finals grade IX p8

August 2, 2021
geometrytangent circlestangential quadrilateral

Problem Statement

A quadrilateral ABCDABCD is circumscribed around a circle ω\omega centered at II. Lines ACAC and BDBD meet at point PP, lines ABAB and CDCD meet at point ££, lines ADAD and BCBC meet at point FF. Point KK on the circumcircle of triangle E1FE1F is such that ∠IKP=90o\angle IKP = 90^o. The ray PKPK meets ω\omega at point QQ. Prove that the circumcircle of triangle EQFEQF touches ω\omega.
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