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Prove line tangent to circle given segment is bisected

Source: CSEMO 2018 Grade 10 Q3

July 30, 2018
geometry

Problem Statement

Let OO be the circumcenter of acute △ABC\triangle ABC(AB<ACAB<AC), the angle bisector of ∠BAC\angle BAC meets BCBC at TT and MM is the midpoint of ATAT. Point PP lies inside △ABC\triangle ABC such that PB⊥PCPB\perp PC. D,ED,E distinct from PP lies on the perpendicular to APAP through PP such that BD=BP,CE=CPBD=BP, CE=CP. If AOAO bisects segment DEDE, prove that AOAO is tangent to the circumcircle of △AMP\triangle AMP.
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