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 .

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