Far Eastern Mathematical Journal

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The variety of generalizations of the Ptolemy's theorem


Astapov N.S., Astapov I.S.

2019, issue 2, Ñ. 129–137


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.

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

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