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Coaxal Circles TheoremThe family of the Apollonian circles defined by two points, say, A and B have a few characteristic properties:
A family of circles that satisfy #2 and #6 is said to form a pencil of circles. The circles are said to be coaxal (sometimes coaxial), because all of them have the same axis of symmetry and any pair has the same radical axis. This property is obviously shared by a family of circles through a given point, say, A and tangent at A to each other. Perhaps, even more interestingly, the same holds for the family of circles that pass through two given points, A and B: their centers lie on the perpendicular bisector of AB; in addition, the radical axis of any pair passes through their common points: A and B. Since any pairs of circles that have the same radical axis are bound to have their centers collinear, sharing the radical axis by any pair of circles in the family is the defining property of the circles. Definition
Coaxal families of circles come in three varieties:
The Coaxal Circles Theorem asserts that the families come out in pairs: all circles in one family is orthogonal to all the circles in the other. The pairing is simple.
Coaxal Circles Theorem
The second part is obvious as of the two perpendicular lines (axes of both families), one is tangent to all the circles in one family, the other is tangent to all the circles in the other family. To prove the first part, consider the inversion t in the unit circle centered at, say, A, the way we did in the discussion on the Apollonian circles. Any circle through A inverts into a straight line. Any circle through A and B inverts into a straight line through t(B). On the other hand, the Apollonian circles invert into the family of circles centered at t(B). These circles and the lines through t(B) are obviously orthogonal. Since, an inversion, as is its inverse, is an angle preserving transformation the original families of circles are also orthogonal. (The applet showing the Apollonian circles provides a clear visualization of this property. Just choose (The families of circles paired via the Coaxal Circles Theorem are known as conjugate.)  The Coaxal Circles Theorem yields an easy proof of the #6 property of the Apollonian circles. Pick a point on the perpendicular bisector of AB and consider circle C from the conjugate family centered at this point. C is orthogonal to any circle from the given Apollonian family. This in particular means that the radius-vector of any of the circles to the point common with C is orthogonal to the corresponding radius-vector of C to the same point. It follows that the latter is tangent to the circle at hand and that the square of its radius is exactly the power of its center with respect to any of the circles in the Apollonian family. References
Copyright © 1996-2008 Alexander Bogomolny |
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