Monge's theorem

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In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is inside one of the others, the three intersection points of the three pairs of external tangent lines are collinear. For any two circles in a plane, an external tangent is a line that is tangent to both circles but does not pass between them. There are two such external tangent lines for any two circles. If two of the circles are of equal sizes, then their two external tangent lines are parallel. If they are considered to intersect at the point at infinity, then the other two intersection points must be on a line passing through the same point at infinity, i.e., it must be parallel to the aforementioned parallel external tangents.

Illustration of Monge's theorem. Circles A, B, and C, determine three collinear points P1, P2, and P3.

[edit] Proofs

Monge's theorem can be proved by using Desargues' theorem.

It also has a "3-dimensional" proof as follows. Consider the three spheres S_A, S_B ,S_C with the circles A, B, C as equators. The point P1 is the vertex of the "smallest" cone containing S_A and S_B, which implies that it is in all planes tangent to both S_A and S_B that do not pass between the spheres. In particular it lies on the line L formed by the intersection of the two planes tangent to all three spheres and not passing between any of them. Similarly the other two points P2 and P3 lie on this line L, so they are collinear.