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# MODULE: Monge's Theorem.

## Part 1: Circles and tangents.

When trying to illustrate the construction for Monge's theorem, we
have to construct tangent lines to several pairs of circles. SketchPad
provides no direct way of doing this, not even a direct construction
of a tangent line to a single circle. The tools provided are more
basic; SketchPad allows only constructions such as a line through two
points, a circle with a given center and radius, and a line
perpendicular to another one passing through a certain point. See the
instructions on SketchPad for a more complete list. Any other
construction should somehow be phrased in terms of these basic ones.
The first task is then, to investigate the properties of tangent lines
to circles.
Using SketchPad, construct a circle, construct a point on it, and a
line through this point. Drag the line around, until it appears to
become tangent to the circle. Is there a special relationship between
the line and the circle? Now draw the segment joining the center of
the circle and the point on it. What angle do the two lines seem to
form? Measure it using the "measure angle" feature. Is this angle
special for the tangent line?

How can we use the previous property to construct a tangent line to a
circle? Construct a circle, a point on it, and construct the tangent
line at that point.

This is all good, but still we don't have a way of constructing a
tangent line passing through a point outside the circle. We would know
how to do it, once we find which point on the circle is the point of
tangency. How do we find it? Look at what we know so far: the
tangent line should be perpendicular to the radius, and it should pass
through the given external point, Q, say. Let's look at all the lines
satisfying these conditions.

Again, construct a circle, and a point Q not on it. Now construct a
pair of lines: one through the center O of the circle, the other
through the point Q, that are perpendicular to each other. Let P be
the point of intersection of the two lines. Now move these lines
around (by dragging point A, the point (other than O) defining one of
the lines), and trace point P. What figure do you seem to get? Make a
conjecture. Construct the figure, and check whether your conjecture
was right or not.

Of all the possible positions of the points P, which one gives you a
point of tangency? Is there more than one? How would you find it
(them?) using the tools available in Sketchpad? And how would you
then construct the tangent lines to a circle through an exterior
point? Can you create a Sketchpad script of this procedure that can
then be used again and again, with any circle and exterior point?

**Next:** *Tangents to circles*

**Up:** *Table of Contents*

**Prev:** *Introduction*

*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: Thu Aug 1 07:39:12 1996