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Fantastic tangency

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 9.4

March 19, 2024
geometrytangency

Problem Statement

Points E,FE, F are selected on sides AC,ABAC, AB respectively of triangle ABCABC with AC=ABAC=AB so that AE=BFAE = BF. Point DD is chosen so that D,AD, A are in the same halfplane with respect to line EFEF, and △DFE∼△ABC\triangle DFE \sim \triangle ABC. Lines EF,BCEF, BC intersect at point KK. Prove that the line DKDK is tangent to the circumscribed circle of △ABC\triangle ABC.
Proposed by Fedir Yudin
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