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Accessible Unlicensed Requires Authentication Published by De Gruyter June 7, 2017

On parametric spaces of bicentric quadrilaterals

Farzali Izadi, Foad Khoshnam, Allan J. MacLeod and Arman Shamsi Zargar
From the journal Mathematica Slovaca

Abstract

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both a circumcircle passing through the four vertices and an incircle having the four sides as tangents. Consider a bicentric quadrilateral with rational sides. We discuss the problem of finding such quadrilaterals where the ratio of the radii of the circumcircle and incircle is rational. We show that this problem can be formulated in terms of a family of elliptic curves given by Ea : y2 = x3 + (a4 − 4a3 − 2a2 − 4a + 1)x2 + 16a4x which have, in general, (ℤ/8ℤ), and in rare cases (ℤ/2ℤ × ℤ/8ℤ) as torsion subgroups. We show the existence of infinitely many elliptic curves Ea of rank at least two with torsion subgroup ℤ/8ℤ, parametrised by the points of an elliptic curve of rank at least one, and give five particular examples of rank 5. We, also, show the existence of a subfamily of Ea whose torsion subgroup is ℤ/2ℤ × ℤ/8ℤ.


(Communicated by Stanislav Jakubec)


Acknowledgement

The authors would like to thank the respected referees for carefully reading of the paper and offering insightful comments, which improved the presentation of the paper.

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Received: 2014-9-18
Accepted: 2016-4-6
Published Online: 2017-6-7
Published in Print: 2017-6-27

© 2017 Mathematical Institute Slovak Academy of Sciences