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collinear wanted, 2 circles and circumucircle related

Source: 2022 IMOC G4 https://artofproblemsolving.com/community/c6h2918249p26069468

September 5, 2022
geometrycollinear

Problem Statement

Let △ABC\vartriangle ABC be an acute triangle with circumcircle Ω\Omega. A line passing through AA perpendicular to BCBC meets Ω\Omega again at DD. Draw two circles ωb\omega_b, ωc\omega_c with B,CB, C as centers and BDBD, CDCD as radii, respectively, and they intersect ABAB, ACAC at E,F,E, F, respectively. Let K≠AK\ne A be the second intersection of (AEF)(AEF) and Ω\Omega, and let ωb\omega_b, ωc\omega_c intersect KBKB, KCKC at P,QP, Q, respectively. The circumcenter of triangle DPQDP Q is OO, prove that K,O,DK, O, D are collinear.
proposed by Li4
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