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MODULE: Monge's Theorem.

Part 0: Introduction.

Consider any two circles in the plane, of different radii. Draw the lines that are tangent to both of them, externally. (There are two kind of common tangent lines: if both circles lie on the same side of the tangent line, it is an "external" tangent; otherwise the tangent is "internal").

Since the radii of these two circles are different, these tangent lines are not parallel; they must then intersect at a point P.

Now suppose we had started with three circles in the plane, all of different radii. There are three pairs of these circles, so we get three pairs of external tangents, and three intersection points.

Sketch (as accurately as you can) the drawing described above; Monge's Theorem gives a general result about this drawing. Try to guess what the result might be.

We will work towards constructing the drawing you just made using Geometer's Sketchpad, a computer program for exploration of plane geometry. Along the way, we will see how the software can be used to enhance discovery of different geometric properties. Through exploration, we will be led to a more informed guess as to the conclusion of Monge's Theorem. Finally, we will use the dilations (a kind of geometric transformations) available in Sketchpad to explore why Monge's Theorem is true.

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Author: Eduardo Tabacman, revised and edited by Evelyn Sander
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Created: Jun 09 1996 --- Last modified: Jun 11 1996