Inscribing an Isosceles Triangle in an Ellipse - I

This is one of a pair of related applets. An isosceles triangle is inscribed in an ellipse with its base perpendicular to one of the axes of the ellipse. How does the area of the triangle vary with the vertex angle opposite the base? You can drag the BLUE dot in the left hand panel to explore a variety of inscribed triangles, You can see an animation by clicking on the icon in the lower left hand corner of one of the windows. Does your analytic solution behave as expected in limiting cases? How is this parametrization of the inscribed isosceles triangle similar to the parametrization by altitude in the companion applet? How does it differ?
Adapted from "Calculus: The Analysis of Functions" by P.D. Taylor