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Prove that line AP is a median of triangle ABD

Source: Iran Third Round MO 1998, Exam 4, P5

October 31, 2010
geometrycircumcirclegeometry proposed

Problem Statement

In a triangle ABCABC, the bisector of angle BACBAC intersects BCBC at DD. The circle Γ\Gamma through AA which is tangent to BCBC at DD meets ACAC again at MM. Line BMBM meets Γ\Gamma again at PP. Prove that line APAP is a median of △ABD.\triangle ABD.
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