Theorem 3.10

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

(x - x₁)² + (y - y₁)² = r ₁²

and

(x - x₂)² + (y - y₂)² = r ₂²

lie on the line whose equation is

if the line joining the centers is horizontal and

otherwise

Proof: First we remove the parentheses

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

Transpose known terms to the right side.

At this point we must distinguish two cases.

Case 1. y₁ = y₂. Our equation becomes

At this point we need to use our assumption that the circles have different centers. In that case we can solve for x.

a vertical line. Since the circles have different centers, the denominator is not zero. 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