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Prove that lines PQ,BC, and MT are concurrent

Source: Saudi Arabia BMO TST Day III Problem 5

August 3, 2014
geometry unsolvedgeometry

Problem Statement

Let ABCABC be a triangle. Circle Ω\Omega passes through points BB and CC. Circle ω\omega is tangent internally to Ω\Omega and also to sides ABAB and ACAC at T, P,T,~ P, and QQ, respectively. Let MM be midpoint of arc BC^\widehat{BC} (containing T) of Ω\Omega. Prove that lines PQ, BC,P Q,~ BC, and MTMT are concurrent.
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