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Length Condition on Circumcenter Implies Tangency

Source: ISL 2022/G4

July 9, 2023
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Problem Statement

Let ABCABC be an acute-angled triangle with AC>ABAC > AB, let OO be its circumcentre, and let DD be a point on the segment BCBC. The line through DD perpendicular to BCBC intersects the lines AO,AC,AO, AC, and ABAB at W,X,W, X, and Y,Y, respectively. The circumcircles of triangles AXYAXY and ABCABC intersect again at Z≠AZ \ne A. Prove that if W≠DW \ne D and OW=OD,OW = OD, then DZDZ is tangent to the circle AXY.AXY.
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