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AP x AQ = AB x AC , circle pasing through B,C of a ABC, angle bisectors

Source: 2014 Saudi Arabia GMO TST day II p1

July 31, 2020
geometrycircleangle bisector

Problem Statement

Let A,B,CA, B,C be colinear points in this order, ω\omega an arbitrary circle passing through BB and CC, and ll an arbitrary line different from BCBC, passing through A and intersecting ω\omega at MM and NN. The bisectors of the angles ∠CMB\angle CMB and ∠CNB\angle CNB intersect BCBC at PP and QQ. Prove that AP⋅AQ=AB⋅ACAP\cdot AQ = AB \cdot AC.
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