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Tangents to a circle and fixed point

Source: China TST 2019 Test 2 Day 1 Q1

March 11, 2019
geometryChinaChina TST2019moving points

Problem Statement

ABAB and ACAC are tangents to a circle ω\omega with center OO at B,CB,C respectively. Point PP is a variable point on minor arc BCBC. The tangent at PP to ω\omega meets AB,ACAB,AC at D,ED,E respectively. AOAO meets BP,CPBP,CP at U,VU,V respectively. The line through PP perpendicular to ABAB intersects DVDV at MM, and the line through PP perpendicular to ACAC intersects EUEU at NN. Prove that as PP varies, MNMN passes through a fixed point.
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