KöMaL 2018 C1479

In a triangle $ABC$, $T$ is an interior point of side $AC$ such that $TA=BC$, and $P$ is an interior point of side $AB$ such that the triangles $CBP$ and $PAT$ are congruent. $Q$ is an interior point of side $BC$ such that $TQ$ is not parallel to $AB$ and triangle $BPQ$ is similar to triangle $TCQ$. Prove that $PT=QT$. The problem was deleted from the contest.