How to find point $G$ in this configuration?

Question

Circle $A$ and circle $B$ intersect at $C$ and $D$. Given that $A = (x_1, 0)$, $B = (x_2, 0)$, and the radii of circle $A$ and circle $B$ are $r_1$ and $r_2$, respectively, then how to find $CG$?

Answer 1

Hint

You know: $ r_1\;,\; r_2 \;,\; AB=d $

let: $AG=x$

Than

$ r_1^2-x^2=CG=r_2^2-(d-x)^2 $

solve this equation for $x$.

Answer 2

Hint:

• Look for right triangles containing $\overline{CG}$
• Use Pythagoras