Tangent - point of contact

Question

I want to find out the co-ordinate of point of contact of tangent to a circle from external point when its center and radius are known. Please Help

Answer 1

Let the external point be $(x,y)$, the center of the circle be $(c_1,c_2)$, and the radius be $r$. Then we can calculate the distance from the point to the center: $$d:=\sqrt{(x-c_1)^2+(y-c_2)^2}$$

Since the tangent point together with the center and the external point form a right triangle, we can now calculate the angle from the horizontal line through the center to the tangent points by adding the angle in the triangle to the angle from the center to the external point: $$\theta_1=\arccos\left(\frac{r}{d}\right)+\arctan\left(\frac{y-c_2}{x-c_1}\right)$$ $$\theta_2=-\arccos\left(\frac{r}{d}\right)+\arctan\left(\frac{y-c_2}{x-c_1}\right)$$

Now to find the tangent points $T_1$ and $T_2$, all we have to is add to the center point: $$T_1=(c_1+r\cos\theta_1,c_2+r\sin\theta_1)$$ $$T_2=(c_1+r\cos\theta_2,c_2+r\sin\theta_2)$$