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AD _|_ BC if KX _|_ AB and KY_|_ AC

Source: V.A. Yasinsky Geometry Olympiad 2023 IX p5 , Ukraine

December 12, 2023
geometryperpendicular bisector

Problem Statement

Let II be the center of the circle inscribed in triangle ABCABC. The inscribed circle is tangent to side BCBC at point KK. Let XX and YY be points on segments BIBI and CICI respectively, such that KX⊥ABKX \perp AB and KY⊥ACKY\perp AC. The circumscribed circle around triangle XYKXYK intersects line BCBC at point DD. Prove that AD⊥BCAD \perp BC.
(Matthew Kurskyi)
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