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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.

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    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.2012-09-05

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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.)