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# Monge's Circle Theorem

Draw three circles in the plane, none of which lies completely inside another, and the common external tangent lines for each pair. Then points of intersection of the three pairs of tangent lines lie on a straight line.

Monge's circle theorem has a three-dimensional analog which states that the apexes of the cones defined by four spheres, taken two at a time, lie in a plane (when the cones are drawn with the spheres on the same side of the apex; Wells 1991).

Circle-Circle Tangents, Circle Tangent Line

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## References

Bogomolny, A. "Three Circles and Common Tangents." http://www.cut-the-knot.org/proofs/threecircles.shtml.Bogomolny, A. "Monge via Desargues." http://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtml.Bogomolny, A. "Monge via Desargues II." http://www.cut-the-knot.org/Curriculum/Geometry/MongeDesargues.shtml.Coxeter, H. S. M. "The Problem of Apollonius." Amer. Math. Monthly 75, 5-15, 1968.Graham, L. A. Problem 62 in Ingenious Mathematical Problems and Methods. New York: Dover, 1959. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 115-117, 1990.Petersen, J. Methods and Theories for the Solution of Problems of Geometrical Constructions, Applied to 410 Problems. London: Sampson Low, Marston, Searle & Rivington, pp. 92-93, 1879.Walker, W. "Monge's Theorem in Many Dimensions." Math. Gaz. 60, 185-188, 1976.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 153-154, 1991.

## Referenced on Wolfram|Alpha

Monge's Circle Theorem

## Cite this as:

Weisstein, Eric W. "Monge's Circle Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MongesCircleTheorem.html