KöMaL 2014 A621

In an acute triangle $ABC$ , the feet of the altitudes are $A_1$, $B_1$ and $C_1$, respectively. The midpoint of the side $BC$ is $F$ . The circle $BCB_1C_1$ intersects the line segment $AA_1$ at point $D$ . Let $T$ be that point on the line segment $DF$ for which the line $BT$ is tangent to the circle $AB_1C_1$. Let the line segment $C_1F$ meet the lines $BD$ and $BT$ at $P$ and $Q$ , respectively. Show that the quadrilateral $DPQT$ has an inscribed circle.