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CP bisects angle C at triangle ABC

Source: AIMO 2, German Pre-TST 2009

July 16, 2011
geometrycircumcircleincentergeometry unsolved

Problem Statement

Let triangle ABCABC be perpendicular at A.A. Let MM be the midpoint of segment BC‾.\overline{BC}. Point DD lies on side AC‾\overline{AC} and satisfies ∣AD∣=∣AM∣.|AD|=|AM|. Let P≠CP \neq C be the intersection of the circumcircle of triangles AMCAMC and BDC.BDC. Prove that CPCP bisects the angle at CC of triangle ABC.ABC.
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