ParaInscribedEllipse

Parallelograms inscribed in Ellipse

Given an ellipse with axes-ratio r = a/b, construct all parallelograms inscribed in the ellipse, find the maximal in area and show that all the maximal are characterized by having their diagonals identical to pairs of conjugate diameters of the ellipse.

A simple method to study this problem is to consider the affinity f, mapping the square IJKL to the rectangle ABCD, which has side ratio |AB|/|BC| = r. The affinity maps then the circumcircle of the square to the ellipse and all the squares inscribed in the circle map to the parallelograms of the ellipse which have conjugate diameters. F.e. RSTQ is mapped in NOPM. Since the squares inscribed in the same circle have the same area and the affinities preserve the area ratios, all the parallelograms with conjugate diameters have the same area. Since affinities preserve also area-ratios the area of the ellipse to the area of one of its maximal parallelograms is pi/2 (as is the are of the circle to its inscribed square). Besides, using the inverse affinity g, map an arbitrary parallelogram UVWX to the corresponding rectangle YZA'B' inscribed in the circle. There is a square QRST with sides parallel to this rectangle, which maps through f to a parallelogram NOPM with sides parallel to UVWX. This proves that all parallelograms inscribed in the ellipse have sides parallel to conjugate diameters. By the preservation of areas-ratios through affinities and the results of MaximalRectangle.html , the area(UVWX)/area(MNOP) = area(YZA'B')/area(QRST) and all the parallelograms inscribed in the ellipse have area smaller than the maximal, which is the area of ABCD.
An interesting question that may arise is the one of finding the locus of N*, when all the corresponding parallelograms UVWX have the same area E < 2*a*b. This is handled in the file ParaInscribedEllipse2.html .
The file MaximalTrianglesInEllipse.html contains an analogous discussion for triangles inscribed in an ellipse.
As explained in the file MaximalRectInEllipse.html the area of ABCD is 2*a*b, where 2*a = |EF| and 2*b = |GH| are the axes of the ellipse. Switch to the selection on contour-tool (press CTRL+2), catch and move points N, N* and watch the corresponding parallelograms inscribed in the ellipse and their counterparts inscribed in the circle (resulting from the parallelograms by applying the inverse affinity g).

Parallelograms inscribed in Ellipse

See Also