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BF_|_BC iff circles omega_1 m $\omega_2 are symmetric wrt L

Source: China Additional TST for IMO 2020, P2

November 16, 2020
geometryright angleequal segmentstangent circles

Problem Statement

Given an isosceles triangle △ABC\triangle ABC, AB=ACAB=AC. A line passes through MM, the midpoint of BCBC, and intersects segment ABAB and ray CACA at DD and EE, respectively. Let FF be a point of MEME such that EF=DMEF=DM, and KK be a point on MDMD. Let Γ1\Gamma_1 be the circle passes through B,D,KB,D,K and Γ2\Gamma_2 be the circle passes through C,E,KC,E,K. Γ1\Gamma_1 and Γ2\Gamma_2 intersect again at L≠KL \neq K. Let ω1\omega_1 and ω2\omega_2 be the circumcircle of △LDE\triangle LDE and △LKM\triangle LKM. Prove that, if ω1\omega_1 and ω2\omega_2 are symmetric wrt LL, then BFBF is perpendicular to BCBC.
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