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).
See also Circle-Circle Tangents ,
Circle Tangent Line
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References Bogomolny, A. "Three Circles and Common Tangents." https://www.cut-the-knot.org/proofs/threecircles.shtml . Bogomolny,
A. "Monge via Desargues." https://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtml . Bogomolny,
A. "Monge via Desargues II." https://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. Numberphile.
"Balls and Cones." Featuring T. Tokieda. Jul. 24, 2018. https://www.youtube.com/watch?v=lubGnk0UZt0 . Ogilvy,
C. S. Excursions
in Geometry. Petersen,
J. Methods
and Theories for the Solution of Problems of Geometrical Constructions, Applied to
410 Problems. Walker, W. "Monge's Theorem in Many Dimensions."
Math. Gaz. 60 , 185-188, 1976. Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry. Referenced on Wolfram|Alpha Monge's Circle Theorem
Cite this as:
Weisstein, Eric W. "Monge's Circle Theorem."
From MathWorld https://mathworld.wolfram.com/MongesCircleTheorem.html
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