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I have a scenario where I have two tangents drawn from two edges (AB and AC) and I have to find the coordinates of the circle center. I have the 2D coordinates of points A, B, C and the values of circle radius r and the angle x. Is there a way to calculate the 2D coordinates of point D from these values?

Edit - I also have the coordinates of the points where the tangents meet AB and AC. (Lets call them P (on AB) and Q (on AC)).

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Thank you.

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You can find the equations of lines $AB: a_1x+b_1y+c_1=0$ and $AC: a_2x+b_2y+c_2=0$. Then, you can use the fact that the distance from $D:(p,q)$ to $AB$ and $AC$ are equal to $r$: $$\frac{|a_1p+b_1q+c_1|}{\sqrt{a_1^2+b_1^2}}=\frac{|a_2p+b_2q+c_2|}{\sqrt{a_2^2+b_2^2}} = r$$ Solve this and you will get the coordinates of four points that are equal distance $r$ with $AB$ and $AC$. Use the angle $x$ to find $D$ and its symmetric point to $A$, and compare with $AB$ and $AC$ to see which of the two points is your point $D$.

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  • $\begingroup$ Thank you for the answer, I made an edit on my original question. Will knowing the tangential points change your answer? $\endgroup$ – Madz May 4 '17 at 8:16
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    $\begingroup$ If you know the tangential points, then you just need to find the intersecting point of the perpendicular lines through $AB, AC$ passing from the tangential points, which makes it much easier. :) $\endgroup$ – Lazy Lee May 4 '17 at 10:09

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