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Dal'nevost. Mat. Zh., 2019, Volume 19, Number 2, Pages 129–137 (Mi dvmg404)  

The variety of generalizations of the Ptolemy's theorem

N. S. Astapovab, I. S. Astapovc

a Novosibirsk State University
b Lavrentyev Institute of Hydrodynamics of Siberian Branch of the Russian Academy of Sciences, Novosibirsk
c Lomonosov Moscow State University, Institute of Mechanics

Abstract: The article examines the metric properties of a tetron. In particular case a tetron is a triangle, flat or spatial quadrangle, and also a tetrahedron. The main theorem is proved about the connection of the lengths of the sides, the magnitudes of the plane angles and the magnitude of the dihedral angle of the tetron is proved. Many remarkable theorems about triangles, quadrangles, and tetrahedra are the corollaries of this theorem. Special attention given to equihedral tetrahedra.

Key words: area of an arbitrary quadrilateral, equihedral tetrahedron, tetron theorem, Bretschneider theorem, Ptolemy's inequality, Brahmagupta's inequality.

Full text: PDF file (128 kB)
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UDC: 514.112
MSC: Primary 52A38; Secondary 52A40
Received: 11.10.2019

Citation: N. S. Astapov, I. S. Astapov, “The variety of generalizations of the Ptolemy's theorem”, Dal'nevost. Mat. Zh., 19:2 (2019), 129–137

Citation in format AMSBIB
\Bibitem{AstAst19}
\by N.~S.~Astapov, I.~S.~Astapov
\paper The variety of generalizations of the Ptolemy's theorem
\jour Dal'nevost. Mat. Zh.
\yr 2019
\vol 19
\issue 2
\pages 129--137
\mathnet{http://mi.mathnet.ru/dvmg404}


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