Point of intersection of two circles with center of one circle lying on another circle

Question

Consider the following configuration:

Two circles with center R and S are shown.Now, suppose I know point R and S and let the radius of both circles be $r_1$ and $r_2$. I need to find point V and U. I know that one way is to solve the equations of both circle and finally getting a quadratic equation and solving it.But sometimes it become too calculative.

Is there another simple approach using geometry and where algebra is minimised?

Answer 1

Let $Z$ be the midpoint of $UV$.
Let $x=RZ,y=ZS$.
By Pythagoras, $r_1^2-r_2^2=x^2-y^2$, and also $x+y=r_1$.
Find $x-y$, then $x$ and $y$.