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Triangle geometry with a point satisfying length condition

Source: Latvian TST for Baltic Way 2022 P10

November 24, 2022
geometrycircumcircle

Problem Statement

Let △ABC\triangle ABC be a triangle satisfying AB<ACAB<AC. Let DD be a point on the segment ACAC such that AB=ADAB=AD. Let then XX be a point on the segment BCBC satisfying BD2=BX⋅BCBD^2=BX\cdot BC. Let the circumcircles of the triangles △XDC\triangle XDC and △ABC\triangle ABC intersect at M≠CM \neq C. Prove that the line MDMD goes through the midpoint of the arc BAC^\widehat{BAC} of the circumcircle of △ABC\triangle ABC.
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