Say we have two circles that intersect at points $X$ and $Y$. We also have a point $A$ on one of those circles, call it the "first circle", with a tangent line $T$ at the point $A$. Lines are extended from $A$ to $X$ and $Y$ until they intersect with the other circle, the "second circle" at points $B$ and $C$ respectively.
My end game is to prove that $BC$ and $T$ are parallel; here's what I have so far:
The tangent $T$ is perpendicular to the radius of the first circle. Extend this radius into a line, call it $K$. The chord $BC$ has a perpendicular bisector that passes through the centre of its circle. Extend this perpendicular bisector into a line, call it $L$.
It's clear that K and L are parallel, but I'm unsure of how to prove this.