- Turn in the following Geometer's Sketchpad sketches.
Part I: Sketch tracing P as A moves (see end of Part I) and a sketch of a tangent to to a circle through an external point.

Part II: Sketch tracing m as the lines containing O1 and O2 move (see end of Part II) and a sketch of an external tangent to two circles.

Part III: Sketch of the three pairs of external tangents to three circles. This should illustrate Monge's Theorem, as in 3.5.

Part IV: Sketch of dilating a polygon by marked ratio.

Part IV: Sketch of the composition of two dilations. - Part I: How do you construct a tangent line to a circle through a
point not on the circle? In this section, you observed a mathematical
fact which makes your construction work. State it explicitly.
- Part II: How do you construct an external tangent to two circles?
In this section, you observed a mathematical fact which makes your
construction work. State it explicitly.
- Part III: State Monge's Theorem.
- Part IV: Given two points A and A', does there exist a dilation
such that A' is the image of A under dilation? If not, what are
conditions for such a dilation to exist? Is such a dilation unique?
Explain.
Given four points A, A', B, and B', does there exist a dilation such that A' is the image of A and B' is the image of B? If not, what are conditions for such a dilation to exist? Is such a dilation unique? Explain.

When are two circles related by a dilation? Is such a dilation unique? Relate this to Part II.

- Based on your observations, you should see that the composition
of two dilations is a third dilation. Explain why the three centers of
dilation are collinear.
- Use the results in the previous question to give a logical
argument justifying Monge's Theorem.

## General Sketchpad questions related to Monge's Theorem.

- Read the Web page
http://www.geom.umn.edu/~banchoff/mongepappus/MP.html
and write a paragraph about a new mathematical idea you learned here.
- Pascal's magic hexagon theorem: Use Sketchpad to make a general
statement about any hexagon inscribed in a circle. Your writeup should
reference a sketch. The following picture should give you a hint.

Author: Evelyn Sander

Comments to:
webmaster@geom.umn.edu

Created: Jun 09 1996 ---
Last modified: Tue Jun 11 15:23:09 1996