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Hard Cyclic Quadrilateral

Source: 2020 RMM Shortlist G3

October 8, 2022
geometrycyclic quadrilateralRMMRMM 2020RMM Shortlist

Problem Statement

In the triangle ABCABC with circumcircle Γ\Gamma, the incircle ω\omega touches sides BC,CABC, CA, and ABAB at D,ED, E, and FF, respectively. The line through DD perpendicular to EFEF meets ω\omega at K≠DK\neq D. Line AKAK meets Γ\Gamma at L≠AL\neq A. Rays KIKI and ILIL meet the circumcircle of triangle BICBIC at Q≠IQ\neq I and P≠IP\neq I, respectively. The circumcircles of triangles KFBKFB and KECKEC meet EFEF at R≠FR\neq F and S≠ES\neq E, respectively. Prove that PQRSPQRS is cyclic.
India, Anant Mugdal
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