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Length of QR is independent of P

Source: Benelux Mathematical Olympiad 2012

April 23, 2012
geometrycircumcirclegeometric transformationreflectiontrapezoidgeometry proposed

Problem Statement

In triangle ABCABC the midpoint of BCBC is called MM. Let PP be a variable interior point of the triangle such that ∠CPM=∠PAB\angle CPM=\angle PAB. Let Γ\Gamma be the circumcircle of triangle ABPABP. The line MPMP intersects Γ\Gamma a second time at QQ. Define RR as the reflection of PP in the tangent to Γ\Gamma at BB. Prove that the length ∣QR∣|QR| is independent of the position of PP inside the triangle.
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