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prove that Q, R, V, and W lie on a circle

Source: 2012 Dutch BxMO / EGMO TST p2

September 19, 2018
geometrycirclescircumcircleConcyclic

Problem Statement

Let △ABC\triangle ABC be a triangle and let XX be a point in the interior of the triangle. The second intersection points of the lines XA,XBXA,XB and XCXC with the circumcircle of △ABC\triangle ABC are P,QP,Q and RR. Let UU be a point on the ray XPXP (these are the points on the line XPXP such that PP and UU lie on the same side of XX). The line through UU parallel to ABAB intersects BQBQ in VV . The line through UU parallel to ACAC intersects CRCR in WW. Prove that Q,R,VQ, R, V , and WW lie on a circle.
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