Find two points on two lines in the plane where the line between the two points go through a third point and are equidistant from that point I have the following situation (see pic below). I have two lines $B$, $C$, in the plane, the intersection point $a$, and a point $p$. I need to find the points $b$ and $c$ along $B$ and $C$ such that the line $A$ goes through $p$ and the two segments on each side of $p$ ($pb$ and $pc$) are of equal length, i.e., so that $l_1 = l_2$.

I have tried setting up various equation systems with three unknowns (the lengths $|bc|$, $|ac|$, and $|ab|$), using the Cosine Law but I only get really complicated algebraic equations when trying to solve them.
Any input would be helpful.

Reflect $C$ in $p$, obtaining reflected line $C'$.  Let $c$ be the intersection of $C'$ and $B$.

