A quick solution can be obtained by solving first the problem in the circle and then passing to the ellipse through an affinity. In the circle it is obvious that the parallelograms must be rectangles and having area E < 2*R^2, R being the radius of the circle. These are rectangles ABCD whose sides are tangent to two circles concentric with the circumscribing circle. Construct the affinity mapping the circle to the ellipse and the images of the two circles. The parallelograms we want are the images of those inscribed in the circle EFGH = f(ABCD) and their sides are tangent to two ellipses, homothetic to the circumscribing ellipse. See the file ParaInscribedEllipse.html for a discussion of related problems.

Switch to the selection on contour-tool (press CTRL+2), catch and move points I, J and watch the corresponding parallelograms inscribed in the circle and their images, inscribed in the ellipse. A question that may arise is the location of those parallelograms EFGH, which have minimal/maximal perimeter. This subject is discussed in the file ParaInscribedEllipse3.html .

ParaInscribedEllipse.html

ParaInscribedEllipse3.html

MaximalTrianglesInEllipse.html

MaximalRectInEllipse.html

Produced with EucliDraw© |