Ian Stewart, one of the mods had a great explanation for this problem which I had copied and saved for my notes. Here is a cut and paste...and again credit goes to Ian for this.
"Imagine the circle is in the co-ordinate plane, centre O at (0,0). You might as well let one of the points A be at (1,0) (you can rotate the circle to get it there if you need to). Consider OA to be the base of our triangle: b=1.
Now, if (c,d) is the third point in the triangle, then the height will be |d|. To get the largest area we need the largest height, and that clearly happens when (c,d) is (0,1) or (0.-1). So the maximum area is 1*1/2 = 1/2."
This is very good solution. My way is more sophisticated. I tried to prove that max happen when the corner of circle-based vertex is 45.