Points $Y,Z$ lie on the smaller arc $BC$ of the circumcircle of an acute triangle $\bigtriangleup ABC$ ($Y$ lies on the smaller arc $BZ$). Let $X$ be a point such that the triangles $\bigtriangleup ABC,\bigtriangleup XYZ$ are similar (in this exact order) with $A,X$ lying on the same side of $YZ$. Lines $XY,XZ$ intersect sides $AB,AC$ at points $E,F$ respectively. Let $K$ be the intersection of lines $BY,CZ$. Prove that one of the intersections of the circumcircles of triangles $\bigtriangleup AEF,\bigtriangleup KBC$ lie on the line $KX$. Proposed by Amirparsa Hosseini Nayeri - Iran
