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New characterisations of bicentric quadrilaterals

Published online by Cambridge University Press:  12 October 2022

Martin Josefsson
Martin Josefsson*
Affiliation:
Västergatan 25d, 285 37 Markaryd, Sweden e-mail: martin.markaryd@hotmail.com
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Extract

A bicentric quadrilateral is a convex quadrilateral that can have both an incircle (it is tangential) and a circumcircle (it is cyclic), see Figure 1. We know of only a dozen characterisations of bicentric quadrilaterals published before. In all of them the starting point is either a tangential or a cyclic quadrilateral, which then must satisfy some condition in order also to be of the other type. Before we proceed to prove seven new such necessary and sufficient conditions for bicentric quadrilaterals, we review one characterisation and one property of tangential quadrilaterals that we will apply later.

Type
Articles
Information
The Mathematical Gazette , Volume 106 , Issue 567 , November 2022 , pp. 414 - 426
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Copyright
© The Authors, 2022. Published by Cambridge University Press on behalf of The Mathematical Association

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References

Josefsson, M., On Pitot’s theorem, Math. Gaz. 103 (July 2019) pp. 333–337.Google Scholar
Durell, C. V. and Robson, A., Advanced trigonometry, Dover (2003).Google Scholar
Josefsson, M., More characterizations of cyclic quadrilaterals, Int. J. Geom. 8(2) (2019) pp. 14–32.Google Scholar
Josefsson, M., On the inradius of a tangential quadrilateral, Forum Geom. 10 (2010) pp. 27–34.Google Scholar
Josefsson, M., The area of a bicentric quadrilateral, Forum Geom. 11 (2011) pp. 155–164.Google Scholar
Covas, M. A., Problem 2209, Crux Math. 24(2) (1998) pp. 112–113.Google Scholar
Josefsson, M., Characterizations of bicentric quadrilaterals, Forum Geom. 10 (2010) pp. 165–173.Google Scholar
2004 All-Russian Olympiad, AoPS Online, available at https://artofproblemsolving.com/community/c5164 Google Scholar
Quadrilateral which is a cyclic quadrilateral and tangent qu, AoPS Online, accessed March 2022 at https://artofproblemsolving.com/community/c6h5510 Google Scholar