EMAT 6680, J. Wilson

## INSTRUCTIONS:

Complete each of the following parts. Part One is merely finishing off the write-ups for the semester. Part Two should be new web page items. Part Three is the course evaluation.

Part One

Review your Write-ups 1 to 12. Revise as you wish to reach the point where you feel that, collectively, the 12 write-ups represent your best work for the course. They are an elctronic portofolio of your work.

Part Two

Complete a Write-up on your Web Page for each of the following investigations. This should be individual work.

A. Bouncing Barney. We discussed this investigation in class. Your challenge now is to prepare a write-up on it, exploring the underlying mathematics ideas and conjectures.

Barney is in the triangular room shown here. He walks from a point on BC parallel to AC. When he reaches AB, he turns and walks parallel to BC. When he reaches AC, he turns and walks parallel to AB. Prove that Barney will eventually return to his starting point. How many times will Barney reach a wall before returning to his starting point? Explore and discuss for various starting points on line BC, including points exterior to segment BC. Discuss and prove any mathematical conjectures you find in the situation.

I assume some GSP sketches and explorations will be useful. A highly regarded write-up will examine the extensions and interpretations of this exploration.

B. Ceva's Theorem. Consider any triangle ABC. Select a point P inside the triangle and draw lines AP, BP, and CP extended to their intersections with the opposite sides in points D, E, and F respectively.

1. Explore (AF)(BD)(EC) and (FB)(DC)(EA) for various triangles and various locations of P.

2. Conjecture? Prove it! (you may need draw some parallel lines to produce some similar triangles, OR, you made need to consider areas of triangles within the figure) Also, it probably helps to consider the ratio

3. In your write-up, after the proof, you might want to discuss how this theorem could be used for proving concurrency of the medians (if P is the centroid), the lines of the altitudes (if P is the orthocenter), the bisectors of the angles (if P is the incenter), or the perpendicular bisectors of the sides (if P is the circumcenter). Concurrency of other special points?

4. Explore a generalization of the result (using lines rather than segments to construct ABC) so that point P can be outside the triangle. Show a working GSP sketch.

Part Three