Inside a quadrilateral $ABCD$ with $AB=BC=CD$, the points $P$ and $Q$ are chosen so that $AP=PB=CQ=QD$. The line through the point $P$ parallel to the diagonal $AC$ intersects the line through the point $Q$ parallel to the diagonal $BD$ at the point $T$. Prove that $BT=CT$. Proposed by Mykhailo Shtandenko