


Paper Title:
Ellipses Inscribed in Parallelograms
Author(s):
A. Horwitz
Penn State University,
25 Yearsley Mill Rd.
Media, PA 19063
U. S. A.
alh4@psu.edu
Abstract:
We prove that there exists a unique ellipse of minimal eccentricity, E_{I}, inscribed in a parallelogram, Ð. We also prove that the smallest nonnegative angle between equal conjugate diameters of $E_{I} equals the smallest nonnegative angle between the diagonals of Ð. We also prove that if E_{M} is the unique ellipse inscribed in a rectangle, R, which is tangent at the midpoints of the sides of R, then E_{M} is the unique ellipse of minimal eccentricity, maximal area, and maximal arc length inscribed in R. Let Ð be any convex quadrilateral. In previous papers, the author proved that there is a unique ellipse of minimal eccentricity, E_{I}, inscribed in Ð, and a unique ellipse, E_{O}, of minimal eccentricity circumscribed about Ð. We defined Ð to be bielliptic if E_{I }and E_{O} have the same eccentricity. In this paper we show that a parallelogram, Ð, is bielliptic if and only if the square of the length of one of the diagonals of Ð equals twice the square of the length of one of the sides of Ð .
Full Text PDF: