I have a circle with the radius R, and coordinates of point A outside of the circle, and the coordinates of the center of the circle. I need to find the exact touching point of the tangent from point A.
I appreciate your help.
Let $C$ be the center of the circle and the point you're looking for be $B$. The tangent $AB$ is perpendicular to the radius $CB$, so triangle $ABC$ is right. Since you know the length $|CB|=R$ and the distance $|AC|$ you can find $|AB|$ using Pythagoras, which gives you a circle centered on $A$ that passes through $B$. Alternatively, you can argue that due to Thales's theorem $B$ lies on the circle whose diameter is $AB$.
In any case, you now know two circles that intersect at $B$. Write down their equations and solve for $x$ and $y$. (There will be two solutions, because there are two tangents to the circle passing through $A$).