Geodesic
The modern generalizations of the Ptolemy's theorem
Matematičeskoe obrazovanie, no. 2 (2020), pp. 18-28
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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.
Mots-clés : tetron
Keywords: Ptolemy's theorem, triangle, flat and spatial quadrilateral, equilateral tetrahedron. 
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N. S. Astapov; I. S. Astapov. The modern generalizations of the Ptolemy's theorem. Matematičeskoe obrazovanie, no. 2 (2020), pp. 18-28. http://geodesic.mathdoc.fr/item/MO_2020_2_a1/

[1] Shteingauz G., Zadachi i razmyshleniya, Mir, M., 1972.

[2] Littsman V., Gde oshibka?, Fizmatgiz, M., 1962

[3] ShibinskiiV.M., Primery i kontrprimery v kurse matematicheskogo analiza, Vyssh. shk., M., 2007

[4] Zadachi i uprazhneniya po matematicheskomu analizu dlya vtuzov, ed. B.P. Demidovicha, Astrel, AST, M., 2001

[5] Demidovich B.P., Sbornik zadach i uprazhneniipo matematicheskomu analizu, Astrel, AST, M., 2002

[6] Filippov A.F., Sbornik zadach po differentsialnym uravneniyam, Nauka, M., 1973

[7] Boyarchuk A.K. Golovach G.P., Spravochnoe posobie po vysshei matematike, v. 5, Differentsialnye uravneniya v primerakh i zadachakh, URSS, M., 1999.

[8] Kokster G.S.M., Greittser S.L., Novye vstrechi s geometriei, Nauka, M., 1978

[9] Prasolov V.V., Zadachi po planimetrii, v. II, Biblioteka matematicheskogo kruzhka, 16, Nauka, M., 1986

[10] Prasolov V.V., Zadachi i teoremy lineinoi algebry, Nauka, Fizmatlit, M., 1996

[11] Astapov N.S., Zhukov A.V., “Zamechatelnyichetyrekh vershinnik”, Kvant, 1996, no. 1, 45—47

[12] ShklyarskiiD.O., Chentsov N.N., Yaglom I.M., Izbrannye zadachi i teoremy planimetrii, Biblioteka matematicheskogo kruzhka, 2, 2, pererabotannoe i dopolnennoe, Nauka, M., 1967

[13] Bakelman I.Ya., Inversiya, Populyarnye lektsii po matematike, 44, Nauka, M., 1966

[14] A.A. Zaslavskii, A. B. Skopenkov, M. B. Skopenkov (red.), Elementy matematiki v zadachakh. Cherez olimpiady i kruzhki — k professii, MTsNMO, M., 2018

[15] Astapov N.S., “Teorema o chetyrekhvershinnike”, Matematicheskoe obrazovanie, 2000, no. 2(13), 22—28

[16] Astapov N.S., Noland N.C., “The Remarkable Tetron”, American Mathematical Monthly, 108:4 (2001), 368—370 | DOI | Zbl

[17] Strashevich S., Brovkin E., Polskie matematicheskie olimpiady, Mir, M., 1978

[18] Prasolov V.V., Sharygin I.F., Zadachi po stereometrii, Biblioteka matematicheskogo kruzhka, 19, Nauka, Fizmatlit, M., 1989

[19] Gashkov S.B., Geometricheskie neravenstva. Putevoditel v zadachakh i teoremakh, Librokom, M., 2014

[20] Yaglom I.M., Geometricheskie preobrazovaniya, v. I, Gos. izd-vo tekhn.-teoret. literatury, M., 1956

[21] Efremov D., Novaya geometriya treugolnika, Tipografiya M. Shpentsera, Odessa, 1902

[22] Shay Gueron, “Two Applications of the Generalized Ptolemy Theorem”, American Mathematical Monthly, 109:4. (2002), 362–370 | DOI | Zbl