Ellipse
The attached file is the same model that many other classmates have discovered. Once you've seen this solution, there is no unseeing it. It is so simple, so elegant. The solution was staring me in the eyes for hours, and yet mesmerized I continued on, lured ever further down by an infinity of circular logic.
You see, I was on the hunt for something beyond the problem, I wanted animation! On my first adventure in Geogebra I stumbled upon a tutorial for using sliders, and I have been searching for a use for them every since. In tackling this problem, I couldn't resist this urge and set to work. My vision was simple: trace the circumference of a circle that rolled from one focus point to another and then back again. In order to create the effect of a shrinking circle as it approached the perpendicular bisector of the focus points I created a circle with the compass set to a line segment with a movable point (so that it could be enlarged/reduced) from the maximum to minimum sizes of the height perpendicular to the focus points.
Next I created a slider that ranged from 0-5. It served as a way to stop the three points that I would be animating. Here was the plan, I just had to synchronize the time it took for the circle to slide from one focus point to the other, with the amount of time it took for the point on the line segment that controlled the size of the moving circle, while still having the point on the outside of the circle rotate around the circle in perfect time. This last point, the one circumnavigating the moving circle had trace turned on and that trace was to be the ellipse.
I spent way too much time working on this without success. I can't decide if I'm messing up my math (I have pages of chicken scratch and pythagorean theorem applications), or if perhaps I have made some faulty assumptions about the relationships between the speed needed to synchronize the 3 points. Despite coming up short of my goal, I was able to make the oval as a solid, whereby I let the off-keelter model run for a while and eventually the trace fills in the entire oval. I am setting this aside for now but will post when I've figured it out.