, 2015

There is given an acute-angled triangle $ABC$ with a point $X$ marked on its altitude starting from $C$. Let $D$ and $E$ be the points on the line $AB$ that satisfy $\sphericalangle DCB=\sphericalangle ACE=90^\circ$. Let $K$ and $L$ be the points on line segments $DX$ and $EX$, respectively, such that $BK=BC$ and $AL=AC$. Let the line $AL$ meet $BK$ and $BC$ at $Q$ and $R$, respectively; finally let the line $BK$ meet $AC$ at $P$. Show that the quadrilateral $CPQR$ has an inscribed circle.
(5 pont)