Part A: Monge's Theorem
- From Part 1: 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.
- From Part 2: 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.
- From Part 3: State Monge's Theorem explicitly.
- From Part 4: 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 the conditions for such a dilation to exist? If so, 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 under dilation? If not, what are conditions for such a dilation to exist? If so, is such a dilation unique? Explain.
- 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.
When are two circles related by a dilation? Is such a dilation unique? Relate this to Part II.
Combine these results to give a logical argument justifying 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.
- Look at the picture below of a hexagon inscribed in a circle, and certain lines associated with it. Does this remind you of Monge's Theorem? Can you conjecture a theorem about such inscribed hexagons? Support your conjecture with a sketch.
Part B: Sketchpad Questions
3. Given two points A and B, what is the set of points P such that AP + BP = a constant K? ("AP" is my shorthand for the distance from point A to point P. Also, K must be greater than the distance AB, or there are no points P.) In this exercise, you use Sketchpad to look at this set and formulate a conjecture about this shape.
Turn in your final sketch. Briefly describe the mathematics of how you constructed it. Write your conjecture for the shape traced in part d. Answer the questions in part e.
Part C: Teaching from the High School Geometry Book