**EMAT 6680, J. Wilson**

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**

Return to **EMAT 6680
Home Page**.