Fantastic Triangles

Friend Miah was in a math competition that had a special kind of triangle. A "fantastic triangle" is one where the third angle is double the first plus the second. I can't find any reference to these fantastic triangles, but wanted to see them. So I made an applet, of course. Then it looked to me like a section of an ellipse. I couldn't tell, so I added an ellipse. Then I wondered about other algebraic constraints so added that. The A+B=C constraint is a right triangle and makes a circle. Are all others an ellipse? If so, can we find the ellipse in terms of the constraints? Maybe they are fantastic!