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 = r12

and

(x - x2)2 + (y - y2)2 = r22

lie on the line whose equation is

if the line joining the centers is horizontal and

otherwise

Proof: First we remove the parentheses

x2 - 2x1x + x12 + y2 - 2y1y + y12 = r12

x2 - 2x2x + x22 + y2 - 2y2y + y22 = r22

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

2(x2 - x1)x + x12 - x22 + 2(y2 - y1)y + y12 - y22 = r12 - r22

Transpose known terms to the right side.

2(x2 - x1)x + 2(y2 - y1)y = x22 - x12 + y22 - y22 + r12 - r22

At this point we must distinguish two cases.

Case 1.   y1 = y2.   Our equation becomes

2(x2 - x1)x = x22 - x12 + r12 - r22

At this point we need to use our assumption that the circles have different centers. In that case   x1   and   x2   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.

2(y2 - y1)y = -2(x2 - x1)x + x22 - x12 + y22 - y22 + r12 - r22

Now divide by the coefficient of the unknown.

This can be simplified a little

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