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Further characterisations of tangential quadrilaterals

Published online by Cambridge University Press:  16 October 2017

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

Tangential quadrilaterals are defined to be quadrilaterals in which a circle can be inscribed that is tangent to all four sides. It is well known and easy to prove that a convex quadrilateral is tangential if, and only if, the angle bisectors of all four vertex angles are concurrent at a point, which is the centre of the inscribed circle (incircle). The most well-known and in problem solving useful characterisation of tangential quadrilaterals is Pitot's theorem, which states that a convex quadrilateral is tangential if and only if its consecutive sides a, b, c, d satisfy the relation a + c = b + d [1, pp. 64-67]. If you want to have more background information about characterisations of tangential quadrilaterals, then we recommend you to check out the lovely papers [2, 3, 4], as well as our previous contributions on the subject [5, 6, 7]. These six papers together include about 30 characterisations that are either proved or reviewed there with references.

Type
Articles
Information
The Mathematical Gazette , Volume 101 , Issue 552 , November 2017 , pp. 401 - 411
Copyright
Copyright © Mathematical Association 2017 

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References

1. Andreescu, T. and Enescu, B., Mathematical Olympiad treasures, Birkhäuser, Boston (2006).Google Scholar
2. Worrall, C., A Journey with circumscribable quadrilaterals, Mathematics Teacher 3 (2004) pp. 192–199.Google Scholar
3. Minculete, N., Characterizations of a tangential quadrilateral, Forum Geom. 9 (2009) pp. 113–118.Google Scholar
4. Hess, A., On a circle containing the incentres of tangential quadrilaterals, Forum Geom. 14 (2014) pp. 389–396.Google Scholar
5. Josefsson, M., More characterizations of tangential quadrilaterals, Forum Geom. 11 (2011) pp. 65–82.Google Scholar
6. Josefsson, M., Similar metric characterizations of tangential and extangential quadrilaterals, Forum Geom. 12 (2012) pp. 63–77.Google Scholar
7. Josefsson, M., Angle and circle characterizations of tangential quadrilaterals, Forum Geom. 14 (2014) pp. 1–13.Google Scholar
8. Ljubenović, L., Condition for a quadrilateral to be tangential, Mathematics Stack Exchange (2013), available at: http://math.stackexchange.com/questions/382416/ Google Scholar
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11. Gutierrez, A., Problem 352: Tangential quadrilateral, incircles, common tangent, circumscribable or tangential quadrilateral, GoGeometry, accessed March 2017 at http://www.gogeometry.com/problem/p352_circle_tangential_quadrilateral.htm Google Scholar
12. Josefsson, M., More characterizations of extangential quadrilaterals, International Journal of Geometry 6 (2) (2016) pp. 62–76.Google Scholar