# 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. Sep 5, 2012 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 $l_1$. Sep 5, 2012 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. Sep 5, 2012 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.)