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Probably simple to solve but I'm a bit stuck. I have given two lines that are tangent to a circlce and the circle must goe through P1 (which is the end of Line 1) and P2 (which is the end of Line2).

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

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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

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