# Find the center of circle given two tangent lines and two points

Probably simple to solve but I'm a bit stuck. I am given two lines that are tangent to a circle and the circle must go through $P_1$ (which is the end of Line 1) and $P_2$ (which is the end of Line 2).

How do I calculate the Center Point of that circle? With given lines and points it should be only one solution.

• Calculate the lines orthogonal to your given lines through the given points. Their intersection is the center. – martini Sep 5 '12 at 15:17
• This is of course over-determined (which implies that there is not always a solution). Construct the angular bisctor(s) of the two lines $l_1$ and $l_2$ (or the middle parallel if they ar parallel) and intersect it with the line orthogonal to $l_1$ through $P_1$ to find the center (or two candidates). We do not need the point $P_2$ at all, only a hint, on which side the circle should touch $l_1$. – Hagen von Eitzen Sep 5 '12 at 15:23
• One thing i know is that the lines will never be parallel and that the circle is on the side of the lines where the angle from l1 to l2 is smaller. – user39558 Sep 5 '12 at 15:39

Following the comment by martini: since every radius of circle is perpendicular to the corresponding tangent line, the center $O$ must be such that $OP_1\perp \ell_1$ and $OP_2\perp \ell_2$. This already determines $O$ as the point of intersection of the perpendiculars to $OP_j$ passing through $P_j$, $j=1,2$.
The solution is unique, if it exists; but it does not exist when $|OP_1|\ne |OP_2|$. (The problem is overdetermined, as Hagen von Eitzen said.)