**Theorem
3.10**: The points
which lie on the intersection of two
circles which have different centers
whose equations are

and

lie on the line whose equation is

if the line joining the centers is horizontal and

otherwise

**Proof**: First we remove the parentheses

*x*^{2} - 2*x*_{1}*x* + *x*_{1}^{2} + *y*^{2} - 2*y*_{1}*y* + *y*_{1}^{2} = *r*_{1}^{2}

*x*^{2} - 2*x*_{2}*x* + *x*_{2}^{2} + *y*^{2} - 2*y*_{2}*y* + *y*_{2}^{2} = *r*_{2}^{2}

If we subtract the second equation from the first, we get

Transpose known terms to the right side.

At this point we must distinguish two cases.

**Case 1**. *y*_{1} = *y*_{2}. Our equation becomes

At this point we need to use our assumption that the
circles have different centers. In
that case *x*_{1 }and *x*_{2 }are different, and we can solve for *x*.

a vertical line. This simplifies to

**Case 2**: The *y-*coordinates are different. In this case we
can solve for *y*. Transpose the *x *term to the other side of the
equation.

Now divide by the coefficient of the unknown.

This can be simplified a little