A point about which inversion of two circles produced concentric circles. Every pair of distinct circles has two limiting points.
The limiting points correspond to the point circles of a coaxal system, and the limiting points of a coaxal system are inverse points with respect to any circle of the system.
To find the limiting point of two circles of radii and with centers separated by a distance , set up a coordinate system centered on the circle of radius and with the other circle centered at . Then the equation for the position of the center of the inverted circles with inversion center ,
(1)
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becomes
(2)
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(3)
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for the first and second circles, respectively. Setting gives
(4)
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and solving using the quadratic equation gives the positions of the limiting points as
(5)
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