The Australian Journal of Mathematical Analysis and Applications


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ISSN 1449-5910  

 

Paper Information

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, EI, inscribed in a parallelogram, . We also prove that the smallest nonnegative angle between equal conjugate diameters of $EI equals the smallest nonnegative angle between the diagonals of . We also prove that if EM is the unique ellipse inscribed in a rectangle, R, which is tangent at the midpoints of the sides of R, then EM is the unique ellipse of minimal eccentricity, maximal area, and maximal arc length inscribed in R. Let be any convex quadrilateral. In previous papers, the author proved that there is a unique ellipse of minimal eccentricity, EI, inscribed in , and a unique ellipse, EO, of minimal eccentricity circumscribed about . We defined to be bielliptic if EI and EO have the same eccentricity. In this paper we show that a parallelogram, , is bielliptic if and only if the square of the length of one of the diagonals of equals twice the square of the length of one of the sides of .

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