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The circle tangent to the perpendicular bisector

Source: 2024 Japan MO P4

February 11, 2024
geometry

Problem Statement

Let ABCABC be an acute triangle with AB<ACAB<AC, circumcenter OO. Let MM be the midpoint of the smaller arc BCBC of the circumcircle of triangle ABCABC. Point DD lies on the extension of side ABAB towards BB such that BD=BMBD=BM. Point EE lies on side ACAC (excluding the endpoints) such that CE=CMCE=CM. Let X(≠A)X(\neq A) be the intersection of circumcircles of triangles ABEABE and ACDACD. Prove that the perpendicular bisector of segment DEDE is tangent to the circumcircle of triangle AOXAOX.
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