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Some Polynomial Conditions for Cyclic Quadrilaterals, Tilted Kites and Other Quadrilaterals

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Abstract

In this paper, we investigate some polynomial conditions that arise from Euclidean geometry. First we study polynomials related to quadrilaterals with supplementary angles, this includes convex cyclic quadrilaterals, as well as certain concave quadrilaterals. Then we consider polynomials associated with quadrilaterals with some equal angles, which include convex and concave tilted kites. Some of the results are proved using Groebner bases computations.

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References

  1. comingstorm (https://stackoverflow.com/users/210211/comingstorm). Convex hull of 4 points. Mathematics Stack Exchange. https://stackoverflow.com/a/2122620 (version: 2018-07-07)

  2. Cors, J.M., Roberts, G.E.: Four-body co-circular central configurations. Nonlinearity 25(2), 343 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, Berlin (2013)

    MATH  Google Scholar 

  4. De Villiers, M.: Some adventures in Euclidean geometry. In: Dynamic Mathematics Learning (2009)

  5. Graumann, G.: Investigating and ordering quadrilaterals and their analogies in space-problem fields with various aspects. ZDM 37(3), 190–198 (2005)

    MathSciNet  Google Scholar 

  6. Josefsson, M.: Properties of tilted kites. Int. J. Geom. 7(1), 87–104 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Kandall, G.A.: Classroom capsules: Euler’s theorem for generalized quadrilaterals. Coll. Math. J. 33(5), 403 (2002)

    Article  Google Scholar 

  8. Knill, O.: Some fundamental theorems in mathematics. arXiv:1807.08416 (2018)

  9. Pech, P.: On equivalence of conditions for a quadrilateral to be cyclic. In: Murgante, B., Gervasi, O., Iglesias, A., Taniar, D., Apduhan, B.O. (eds.) Computational Science and Its Applications—ICCSA 2011, pp. 399–411. Springer, Berlin (2011)

    Chapter  Google Scholar 

  10. Santoprete, M.: Planarity conditions and four-body central configurations equations with angles as coordinates. J. Geom. Phys. 140, 74–84 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Santoprete, M.: On the uniqueness of co-circular four body central configurations. Arch. Ration. Mech. Anal. 240(2), 971–985 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Santoprete, M.: On the uniqueness of trapezoidal four-body central configurations. Nonlinearity 34(1), 424 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author wish to thank Martin Josefsson for his helpful comments and suggestions.

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  1. Department of Mathematics, Wilfrid Laurier University, Waterloo, Canada

    Manuele Santoprete

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  1. Manuele Santoprete
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Correspondence to Manuele Santoprete.

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Santoprete, M. Some Polynomial Conditions for Cyclic Quadrilaterals, Tilted Kites and Other Quadrilaterals. Math.Comput.Sci. 17, 24 (2023). https://doi.org/10.1007/s11786-023-00574-7

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  • DOI: https://doi.org/10.1007/s11786-023-00574-7

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