Problem

Source: Cono Sur 2021 #6

Tags: geometry



Let ABC be a scalene triangle with circle Γ. Let P,Q,R,S distinct points on the BC side, in that order, such that BAP=CAS and BAQ=CAR. Let U,V,W,Z be the intersections, distinct from A, of the AP,AQ,AR and AS with Γ, respectively. Let X=UQSW, Y=PVZR, T=URVS and K=PWZQ. Suppose that the points M and N are well determined, such that M=KXTY and N=TXKY. Show that M,N,A are collinear.