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Mat. Sb., 2003, Volume 194, Number 3, Pages 3–24 (Mi msb718)  

This article is cited in 14 scientific papers (total in 14 papers)

Inscribed polygons and Heron polynomials

V. V. Varfolomeev

Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University)

Abstract: Heron's well-known formula expressing the area of a triangle in terms of the lengths of its sides is generalized in the following sense to polygons inscribed in a circle: it is proved that the area is an algebraic function of the lengths of the edges of the polygon. Similar results are proved for the diagonals and the radius of the circumscribed circle. The resulting algebraic equations are studied and elementary geometric applications of the algebraic results obtained are presented.

DOI: https://doi.org/10.4213/sm718

Full text: PDF file (308 kB)
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English version:
Sbornik: Mathematics, 2003, 194:3, 311–331

Bibliographic databases:

UDC: 513.7
MSC: 51M25, 52A10, 52A38
Received: 04.03.2002

Citation: V. V. Varfolomeev, “Inscribed polygons and Heron polynomials”, Mat. Sb., 194:3 (2003), 3–24; Sb. Math., 194:3 (2003), 311–331

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. V. Varfolomeev, “Galois groups of the Heron–Sabitov polynomials for inscribed pentagons”, Sb. Math., 195:2 (2004), 149–162  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Maley F.M., Robbins D.P., Roskies J., “On the areas of cyclic and semicyclic polygons”, Adv. in Appl. Math., 34:4 (2005), 669–689  crossref  mathscinet  zmath  isi
    3. Pak I., “The area of cyclic polygons: Recent progress on Robbins' conjectures”, Adv. in Appl. Math., 34:4 (2005), 690–696  crossref  mathscinet  zmath  isi
    4. Fedorchuk M., Pak I., “Rigidity and polynomial invariants of convex polytopes”, Duke Math. J., 129:2 (2005), 371–404  crossref  mathscinet  zmath  isi  elib
    5. J. Math. Sci. (N. Y.), 158:6 (2009), 899–903  mathnet  crossref  zmath  elib
    6. G.. Khimshiashvili, “Configuration spaces and signature formulas”, J. Math. Sci. (N. Y.), 160:6 (2009), 727–804  crossref  mathscinet  zmath
    7. Connelly R., “Comments on generalized Heron polynomials and Robbins' conjectures”, Discrete Mathematics, 309:12 (2009), 4192–4196  crossref  mathscinet  zmath  isi  elib
    8. Guo R., Sonmez N., “Cyclic Polygons in Classical Geometry”, C R Acad Bulgare Sci, 64:2 (2011), 185–194  mathscinet  zmath  isi
    9. A. D. Mednykh, “Brahmagupta formula for cyclic quadrilaterals in the hyperbolic plane”, Sib. elektron. matem. izv., 9 (2012), 247–255  mathnet
    10. A. M. Zhukova, “Nabory vpisannykh konfiguratsii tipichnykh sharnirnykh pyatiugolnikov”, Tr. SPIIRAN, 21 (2012), 203–227  mathnet
    11. Panina G.Yu., Khimshiashvili G.N., “On the Area of a Polygonal Linkage”, Dokl. Math., 85:1 (2012), 120–121  crossref  mathscinet  zmath  isi  elib  elib
    12. G. Khimshiashvili, D. Siersma, “Critical configurations of planar multiple penduli”, J Math Sci, 2013  crossref  mathscinet  elib
    13. Abrams A., Pommersheim J., “Spaces of Polygonal Triangulations and Monsky Polynomials”, Discret. Comput. Geom., 51:1 (2014), 132–160  crossref  mathscinet  zmath  isi
    14. Abrosimov N., Mednykh A., “Area and Volume in Non-Euclidean Geometry”, Eighteen Essays in Non-Euclidean Geometry, Irma Lectures in Mathematics and Theoretical Physics, 29, eds. Alberge V., Papadopoulos A., European Mathematical Soc, 2019, 151–189  mathscinet  zmath  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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