state and prove monge's theorem Share with your friends Share 0 Ambilimon answered this Dear Student, Please find below the solution to the asked query: Monge's theorem states that for three circles on a plane, none of them inside another, the three points of intersection of the three pairs of common tangents are collinear.The three dimensional proof of the theorem goes like this:Consider three spheres resting on a plane like three marbles of different sizes on the floor.Let us name the plane as Plane 1.Plane 1 is touching the spheres below.Now consider another plane which is touching the plane above.Let us call this plane as Plane 2.It may be imagined as these two planes forming a V shaped space and the spheres fitting inside this space.Plane 1 and 2 will intersect along a line.We name this line of intersection as I.Now consider any two of the three spheres.It is possible for the two spheres to fit into a single cone.In other words, there is a cone which is tangential to each pair of spheres.We have three spheres, hence three different pairs of spheres, and thus three cones.Now consider the three cones and the planes.The cones are tangential to pairs of sheres, and the planes are tangential to all the spheres.Since the planes meet at I, the apex of each of the three cones must also be on I.To bring this down to two dimensions, think of a new plane, Plane 3, that passes through the centres of the three spheres.Planes 1, 2 and 3 intersect at I.Plane 3 intersects the spheres as circles.The cones would appear as lines tangential to pair of circles meeting at I, for each pair of circles.This completes the proof. If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible. 1 View Full Answer
