**Next:** *Tangents to circles*

**Up:** *Table of Contents*

**Prev:** *Introduction*

# MODULE: Monge's Theorem.

## Part 1: Circles and tangents.

Monge's theorem involves lines which are externally tangent to a pair
of circles, so to illustrate it in Sketchpad we must be able to
construct such a line. Sketchpad provides no direct way of doing
this, or even of directly constructing any tangent line to a single
circle. The tools provided are more basic: Sketchpad constructs
objects like the line through two points, the circle with a given
center and radius, and the line perpendicular to a given one passing
through a certain point. Essentially, the tools of Geometer's Sketchpad are
the straight-edge and compass, along with some built in constructions
from Euclidean geometry. Any other construction should somehow be
built in terms of these basic ones. Thus, the first task is to
investigate the properties of tangent lines to circles, in order to
find such a construction.
Using Sketchpad, construct a circle, 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, and a point on it, and then construct the tangent
line at that point.

This is all good, but suppose we are given a circle centered at O,
and a point Q outside the circle. There are two lines through Q
which are tangent to the circle, and we want to be able to construct (one
of) them. If we could find which point P on the circle is the point of
tangency, then we could construct the line as before. How do we find P?
Look at what we know so far: the
tangent line should be perpendicular to the radius OP, and it should pass
through the given external point Q. Let's look at all the lines
satisfying these conditions.

So, construct a circle, and a point Q not on it. Now construct a
line through the center O of the circle to some point A, and construct
a perpendicular line to it through Q. Let P be
the point of intersection of these two perpendicular lines.
Now drag point A around, moving the whole configuration, 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 these possible positions of the points P, which makes line PQ
tangent to the circle? Is there more than one? How would you find it
(them?) using the tools available in Sketchpad?
Putting this all together, given a circle and an exterior point Q, how
do you construct the line through Q tangent to the circle?
Create a Sketchpad script of this procedure that can then be
used again and again, with any circle and exterior point. You may want
to hide the figures used in the construction so that only
the tangent line appears.

**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: Jun 11 1996