I found a simplistic way to find the triangle area, that does not involve piecewise functions. You basically use the angle measured from the fixed vertex in your calculation of the area. The resulting equation is still somewhat hairy and spikey, but at least it's all in one piece.
Here is the demo that illustrates the moving triangle, with only one fixed point and two vertices rotating each at a different rate around the circle.
Screenshot (much less exciting than the animation linked to above):
Graph of area over time. Notice that this is much more interesting, because you can see the times at which the two points overlap on the circle. NOTE: This graph is incorrect; corrected version is below in the Addendum!
Now, here is my question after looking at this problem for a bit and feeling like it's driving me nuts: Why are there 9 periods between t = 0 and t = 20 (by examination of the graph)? Is there a mathematical way of figuring this out? (Is it simply that one point completes 4 cycles and the other completes 5 cycles in 20 seconds, so the sum is how many times both points are collinear with E?? If so, how can you justify that geometrically?)
ARGH. I must be thinking about this the roundabout way, and I hate that! If you can see through this fog, please enlighten me. Otherwise it'll keep driving me nuts.
Addendum 7/31/10: Thanks for the catch, Matt! Here's the fixed graph.
This time, I saved the file with all the helper functions for you to look at. I tried to name them descriptively, but ED(x) and EC(x) represent the lengths of two sides as a function of time, and thetaE(x) represents the angle at E as a function of time. AREA(x) is the final area calculated as a function of time, using Law of Sine. Cheers!