# Monge's Theorem

## Circles and Tangents

Click here to view problem number 1. To construct a tangent line to a circle through a point not on the circle, follow my script.

The observed mathematical fact is that a tangent line to a circle is perpendicular to the radius.

## External Tangents to Two Circles

To construct an external tangent to two circles follow this script.

## Monge's Theorem

Monge's Theorem: Given three circles of different radii, the centers of symmetry of the three pairs of circles are collinear.

## Dilations

Given two points A and A', there does exist a dilation such that A' is the image of A.

The sketch of dilating a polygon (circle) by a marked ratio.

A sketch of a composition of two dilations.

## Learning Something New about Monge's Theroem

After investigating Monge's Theorem and reading about Desargures' Theorem, we learned that they are closely related. Desargues' theorem says that if two triangles have their corresponding vertices on three parallel lines, then the intersections of the lines containing the corresponding sides will be collinear. This is related to Monge's theorem that 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. Looking farther into Monge's Theorem we learned that it can be extended into spheres and that the spheres create six points and the lines determined by these points are coplaner.