I have a circle of radius r.
Given two lines tangent to the circle at points (x1,y1) and (x2,y2),
What are the coordinates of the point where the two tangents cross?
Let $(x_m,y_m)$ be the middle of $(x_1,y_1)$ and $(x_2,y_2)$. It turns out that the point $(x_p,y_p)$ you are looking for is a multiple of $(x_m,y_m)$ and that its distance from the origin is $r$ over the distance of $(x_m,y_m)$ from the origin.
Equation of tangent on circle $x^2+y^2=r^2$ at point $(x_1,y_1)$ is $$xx_1+yy_1=r^2$$
Similarly, tangent at $(x_2,y_2)$ is $$xx_2+yy_2=r^2$$
These are two linear equations which can be solved easily for intersection point