Let $ABC$ be an isosceles triangle such that $|AB|=|AC|$. Let $T$ be a point such that $AT||BC$ and $|BT|<|CT|$. With $M$ being the midpoint of $BC$, the rays $TB$ and $TC$ are drawn ($T$ is the starting point of both rays). Points $P$ and $Q$ are chosen off of lines $BT$ and $CT$ respectively such that $|\angle TPM|=|\angle TQM|$ and $|TP|>|TB|$ and $|TQ|>|TC|$. Prove that $ATPQ$ is cyclic.
THE WORK: So far, what I have realized is that $|\angle TAB|=|\angle ABC|=|\angle ACB|$ and that since triangles $TPM$ and $TQM$ share a common side and the angles above that side are equal, two congruent circles can be constructed with a chord $TM$. I am lost here.