KöMaL 2014 B4674

On the circumscribed circle of triangle $ABC$ , a point $X$ is moving along the arc $AC$ not containing vertex $B$ . Let $Y$ and $Z$ denote the points on the extensions of side $BA$ beyond $A$ and side $BC$ beyond $C$ , respectively, for which $AY=AX$ and $CZ=CX$ . What is the locus of the midpoint of line segment $YZ$ ?