Prev: Dilations

# Monge's Theorem Homework

Here are some questions from the lab on Monge's Theorem.
1. 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.

2. 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.

3. 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.

4. Part III: State Monge's Theorem.

5. 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.

6. 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.

7. Use the results in the previous question to give a logical argument justifying Monge's Theorem.

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

8. 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.

9. 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.