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Hard geometry with concurrency on OI

Source: IMEO Problem 6

July 15, 2020
IMEOgeometryconcurrency

Problem Statement

Let OO, II, and ω\omega be the circumcenter, the incenter, and the incircle of nonequilateral △ABC\triangle ABC. Let ωA\omega_A be the unique circle tangent to ABAB and ACAC, such that the common chord of ωA\omega_A and ω\omega passes through the center of ωA\omega_A . Let OAO_A be the center of ωA\omega_A. Define ωB,OB,ωC,OC\omega_B, O_B, \omega_C, O_C similarly. If ω\omega touches BCBC, CACA, ABAB at DD, EE, FF respectively, prove that the perpendiculars from DD, EE, FF to OBOC,OCOA,OAOBO_BO_C , O_CO_A , O_AO_B are concurrent on the line OIOI.
Pitchayut Saengrungkongka
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