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Triangle configuration with line intersection on circumcircle

Source: 2018 Latvia BW TST P9

March 26, 2022
geometrycircumcircle

Problem Statement

Acute triangle △ABC\triangle ABC with AB<ACAB<AC, circumcircle Γ\Gamma and circumcenter OO is given. Midpoint of side ABAB is DD. Point EE is chosen on side ACAC so that BE=CEBE=CE. Circumcircle of triangle BDEBDE intersects Γ\Gamma at point FF (different from point BB). Point KK is chosen on line AOAO satisfying BK⊥AOBK \perp AO (points AA and KK lie in different half-planes with respect to line BEBE). Prove that the intersection of lines DFDF and CKCK lies on Γ\Gamma.
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