Theorem 3.10: The points which lie on the intersection of two circles which have different centers whose equations are
lie on the line whose equation is
if the line joining the centers is horizontal and
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
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 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.
Now divide by the coefficient of the unknown.
This can be simplified a little
next theorem (3.11)