**Next:** *Monge's Theorem*

**Up:** *Table of Contents*

**Prev:** *Circles and tangents*

# MODULE: Monge's Theorem.

## Part 2: Tangents to circles.

We have found how to construct tangent lines to circles, passing through a
given point either on the circle or outside of it. We now want to build
on this to construct the lines tangent to two circles (of
different radii) at the same time, since this is what we need for Monge's
Theorem.
Draw two circles, and pick any point P on one of them. Construct the
line through this point that is tangent to the other circle, using
what we learned in Part I. Now drag the point P around, until the
line looks tangent to both circles. Can you guess where the other
external tangent will be? Can you get it by using the reflection
command? Where is the mirror line?

Now, given this common tangent line, draw the radii of the two circles to
the points of tangency. What angle do the radii and tangent line form?
Why is that? What is true about these two radii?

The common tangent to the two circles passes then through points that
have parallel radii. Let's explore this a bit more closely. Draw two
circles, and any line l through the center of one of them (i.e., any
line containing a radius). Now construct a parallel line through the
center of the other circle. Each line intersects its circle in two
opposite points. Pick the upper one (or rightmost one, or whatever)
on each circle, and construct the line m through them.
We know that for some particular
choice of the first line l, this constructed line m would be
the common tangent we are looking for.
Move the line l without moving the circles, and use the
"trace line" option on the Display menu to trace the locus of all the
lines m you get. Do you notice anything peculiar? Can you check
this? Do you see how can you construct it?

As we mentioned before, the common
external tangent is one of the lines whose locus we traced above. So
it too must pass through their common point, the so-called "Center of
Symmetry" of the two circles. How can you use this fact to construct
the common tangent lines of two circles? Can you create a script for
this procedure?

Notice that when we found the center of symmetry, we picked the two
upper points on each circle. What happens when we pick an upper point
on one of the circles, and a lower one on the other circle? What do
we get if we repeat the construction for the external tangents, but
using these points now?

**Next:** *Monge's Theorem*

**Up:** *Table of Contents*

**Prev:** *Circles and tangents*

*The Geometry Center Home Page*
Author: Eduardo Tabacman, revised and edited by Evelyn Sander

Comments to:
webmaster@geom.umn.edu

Created: Jun 09 1996 ---
Last modified: Jun 11 1996