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Variable point on the median

Source: APMO 2019 P3

June 11, 2019
geometryAPMO

Problem Statement

Let ABCABC be a scalene triangle with circumcircle Γ\Gamma. Let MM be the midpoint of BCBC. A variable point PP is selected in the line segment AMAM. The circumcircles of triangles BPMBPM and CPMCPM intersect Γ\Gamma again at points DD and EE, respectively. The lines DPDP and EPEP intersect (a second time) the circumcircles to triangles CPMCPM and BPMBPM at XX and YY, respectively. Prove that as PP varies, the circumcircle of △AXY\triangle AXY passes through a fixed point TT distinct from AA.
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