# Find coordinates of circle center

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)).

Thank you.

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 $D$.
• 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. :) – Lazy Lee May 4 '17 at 10:09