Lesson 4: Putting it all together

Basics

This session presents you with a series of problems from different areas of geometry. You will have a chance to model these problems with Sketchpad and to at least make conjectures for some of their solutions.
During the last hour of the session you will each have a chance to present some of the results of your investigations into these problems. We will conclude with a discussion of your thoughts on the applicability of these techniques and ideas in the classroom.

Problems

1. Inscirbe a triangle ABC in a circle. Construct the orthocenter (intersection of the altitudes) for this triangle. You have a script for this. Determine the locus of this orthocenter as A is moved around the circle. Verify your conjecture with Sketchpad.
2. Repeat the previous problem, using the center of the inscribed circle instead of the orthocenter.
3. Given three parallel lines, construct an equilateral triangle with one vertex on each line.
4. Construct four points A, B, C, and D. For each line m through A, construct the three rectangles whose sides go through B, C, and D and are perpendicular or parallel to m. Construct the centers X, Y, and Z of these rectangles. Find the loci of the points X, Y, and Z as m varies among all lines through A. Verify your conjecture with Sketchpad.
5. Construct a circle, a point A on it, and a point B outside it. Find the locus of the midpoint, M, of the segment AB as A moves around the circle. Verify your conjecture with Sketchpad.
6. Bonus Problem Construct two circles, one outside the other, a point A on one of the circles and a point B on the other. Find the locus of the midpoint, M, of segment AB as A and B move around their respective circles. Verify your conjecture with Sketchpad. This is a problem from the 1996 Putnam Contest, the premiere undergraduate mathematics competition in North America!