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prove that 4 points lie on the same circle

Source: Brazilian Mathematical Olympiad 2024, Level 2, Problem 2

October 12, 2024
geometryTrianglemidpointscongruent trianglescircumcircleparallel

Problem Statement

Let ABC ABC be a scalene triangle. Let E E and F F be the midpoints of sides AC AC and AB AB , respectively, and let D D be any point on segment BC BC . The circumcircles of triangles BDF BDF and CDE CDE intersect line EF EF at points K≠F K \neq F , and L≠E L \neq E , respectively, and intersect at points X≠D X \neq D . The point Y Y is on line DX DX such that AY AY is parallel to BC BC . Prove that points K K , L L , X X , and Y Y lie on the same circle.
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