Question

Two circles intersect at A and B. The tangent at B to one circle meets the second again at D, and a straight line through A meets the first circle at P and the second at Q. Prove that BP is parallel to DQ

Answer 1

Tow circles intersect at $A$ and $B$ The tangent at $B$ to one circle meets the second again at $D,$ $A$ straight line through $A$ meets the first circle at $P$ and the second at $Q$.
Let $O$ be the center of a circle and $BP$ $\&$ $DQ$ be two chords.
Let $BD$ be the diameter bisecting $BP$ and $DQ$ at $L$ and $M$ respectively.
So, $OL\perp BP=\angle BLO=90$
Similarly $\angle DMO=90°\angle BLO=\angle DMO$
But these are alternate angles, hence $BP||DQ$