I have a geometry problem that asks to prove that 2 lines intersect on the circumference of a circle. It goes like this:
There are 2 circles, $O_1$ and $O_2$with centres $A$ and $B$ respectively, with the $A$ lying on the circumference of $O_2$. A point $P$ is chosen on $O_2$ so that it isn't in $O_1$. A line tangent to $O_1$ through $P$ meet $O_1$ at $S$, and it intersects $O_2$ again in $Q,$ with $Q$ and $P$ lying on the same side of $AB$. A line through $Q$ is tangent to $O_1$ again at $T$. A point $M$ is the foot of the perpendicular from $P$ to $AB$. Prove that $MT$ intersects $PS$ at $S$.
I tried to use tan-chord theorem and joining the intersections of $O_1$ and $O_2$ and joining $A$ and $B$ with the points of tangency, and then finding equal angles using isosceles triangles, but I couldn't get anywhere from there.
Is there a general way to prove that 2 lines intersect each other on the circumference?