ISL 1999 G4

For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$ , the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ( $CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$ . We denote by $\Sigma$ the set of all triangles $T$ for which $\widehat{XYZ} = 45$ . Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.

Attachments: