In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear.

Monge's Theorem is a theorem in Euclidean geometry. Given three circles, and , none of which lies completely inside one of the others, if we construct the intersections of their common external tangents, these intersections are collinear.

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).

Monge's theorem says that for three disjoint circles of unequal radii, with no one contained in any other, the pairs of external tangents meet in three points that are collinear.

Theorem (my conjecture) If we extend opposite sides of a hexagon inscribed in a circle, those sides will meet in three distinct points, and those points will lie on a line.

Monge's Theorem Monge:=proc () local d1,d2,d3,C1,C2,C3,R1,R2,R3,a,b: d1:=d3*sin (a)/sin (a+b): d2:=d3*sin (b)/sin (a+b): C1:= [0,0]: C2:= [d3,0]: C3:= [d2*cos (a),d2*sin (a)]: Colinear ( CommonTangentsMeetingPt (C1,R1,a,d2,R3) , CommonTangentsMeetingPt (C3,R3,-b,d1,R2) , CommonTangentsMeetingPt (C1,R1,0,d3,R2) ): end: Previous Definitions ...

The diagram below shows an example of Monge's theorem. Each pair of circles in the plane defines a pair of external common tangents. In the diagram below, we have three circles, and therefore three pairs of common tangents, coloured red, blue and green. Let X X $X$ be the point of intersection of the common tangents for the circles centred at A A

In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear.

6) We learned that Monge’s theorem can be applied to three dimensions. In two dimensions we already seen that the external tangents of three non-uniform spheres lie on a line. Likewise, the web site showed that if we add one more sphere to the problem the external tangents will all lie on a plane.

The setup for Monge’s circle theorem consists of three random circles drawn in the plane. Well, to be honest, they can’t be entirely random – we can’t allow a circle that is entirely inside another circle.