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# MODULE: Monge's Theorem.

## Part 2: Tangents to circles.

We know how to construct tangent lines to circles, passing through a
given point; a point either on the circle or outside of it. We still
must determine how to construct the lines tangent to two circles (of
different radii) at the same time.
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? Now draw the radii of the two circles at their respective
points of tangency. What angle do the radii and tangent line seem to
form? Why is that?

The common tangent to the two circles passes then through points that
give 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. These lines intersect each corresponding
circles in two points. Pick the upper ones on each circle, and
construct the line m through them. We know that for a particular
choice of the first line, l, we would get 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 line? 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

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Created: Jun 09 1996 ---
Last modified: Thu Aug 1 07:39:29 1996