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Easy for a #6 -- Prove AP and AQ are isogonal

Source: Taiwan 2014 TST2, Problem 6

July 18, 2014
geometrycircumcirclegeometric transformationsimilar trianglesgeometry proposed

Problem Statement

Let PP be a point inside triangle ABCABC, and suppose lines APAP, BPBP, CPCP meet the circumcircle again at TT, SS, RR (here T≠AT \neq A, S≠BS \neq B, R≠CR \neq C). Let UU be any point in the interior of PTPT. A line through UU parallel to ABAB meets CRCR at WW, and the line through UU parallel to ACAC meets BSBS again at VV. Finally, the line through BB parallel to CPCP and the line through CC parallel to BPBP intersect at point QQ. Given that RSRS and VWVW are parallel, prove that ∠CAP=∠BAQ\angle CAP = \angle BAQ.
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