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On parametric spaces of bicentric quadrilaterals

In: Mathematica Slovaca, vol. 67, no. 3
Farzali Izadi - Foad Khoshnam - Allan J. Macleod - Arman Shamsi Zargar

Details:

Year, pages: 2017, 611 - 622
Keywords:
bicentric quadrilateral, elliptic curve, rank, torsion group
About article:
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, (\mathbb Z/8\mathbb Z), and in rare cases (\mathbb Z/2\mathbb Z× \mathbb Z/8\mathbb Z) as torsion subgroups. We show the existence of infinitely many elliptic curves $Ea$ of rank at least two with torsion subgroup $\mathbb Z/8\mathbb Z$, 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 $\mathbb Z/2\mathbb Z×\mathbb Z/8\mathbb Z$.
How to cite:
ISO 690:
Izadi, F., Khoshnam, F., Macleod, A., Zargar, A. 2017. On parametric spaces of bicentric quadrilaterals. In Mathematica Slovaca, vol. 67, no.3, pp. 611-622. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0295

APA:
Izadi, F., Khoshnam, F., Macleod, A., Zargar, A. (2017). On parametric spaces of bicentric quadrilaterals. Mathematica Slovaca, 67(3), 611-622. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0295
About edition:
Published: 27. 6. 2017