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Nine Circles Theorem


NineCirclesTheorem

Let A, B, and C be three circles in the plane, and let X be any circle touching B and C. Then build up a chain of circles such that Y:CAX, Z:ABY, X^':BCZ, Y^':CAX^', Z^':ABY^', X^(''):BCZ^', where C:C_1C_2C_3 denotes a circle C tangent to circles C_1, C_2, and C_3. Although there are a number of choices for each successive tangent circle in the chain, if the choice at each stage is made appropriately, then the ninth and final circle X^('') coincides with the first circle X (Evelyn et al. 1974, p. 58).


See also

Circle, Six Circles Theorem, Seven Circles Theorem, Tangent Circles

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References

Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Nine Circles Theorem." §3.4 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 58-68, 1974.Tyrrell, J. A. and Powell, M. T. "A Theorem in Circle Geometry." Bull. London Math. Soc. 3, 70-74, 1971.

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Nine Circles Theorem

Cite this as:

Weisstein, Eric W. "Nine Circles Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NineCirclesTheorem.html

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