MathNet 2015-part-01 — 191 problems (EN–VI) Question 108

Let the incircle of the triangle $ABC$ touch the side $AB$ at point $Q$; the incircles of the triangles $QAC$ and $QBC$ touch $AQ$, $AC$ and $BQ$, $BC$ at points $P$, $T$ and $D$, $F$, respectively.
Prove that $PDFT$ is a cyclic quadrilateral.