1683-3414 1814-0807 Владикавказский математический журнал Vladikavkaz Mathematical Journal Южный математический институт - филиал Федерального государственного бюджетного учреждения науки Федерального научного центра «Владикавказский научный центр Российской академии наук» (ЮМИ ВНЦ РАН) One Application of Ptolemy's Theorem Одно применение теоремы Птолемея 10.46698/o1944-0990-7792-e 17994 12 2025 27 4 103 108 https://vmj.ru/eng/archive/detail.php?ELEMENT_ID=18027&SECTION_ID=647 Никоноров Ю. Г. Nikonorov Yu. G. nikonorov2006@mail.ru Оскорбин Д. Н. Oskorbin D. N. oskorbin.dn@phystech.edu Южный математический институт - филиал ВНЦ РАН, Россия, 362025, Владикавказ, ул. Ватутина, 53 South Mathematical Institute of VSC RAS, 53 Vatutina St., Vladikavkaz 362025, Russia Московский физико-технический институт, Россия, 141701, Долгопрудный, Институтский переулок, 9 Moscow Institute of Physics and Technology, 9 Institutsky Per., Dolgoprudny 141701, Russia In this paper, we present a simple proof of one result from a recent paper by Yu. G. Nikonorov and O. Yu. Nikonorova, the original proof of which is rather cumbersome and based on the study of polynomial ideals and symbolic computations using a computer. The new proof is based on Ptolemy's theorem on inscribed quadrilaterals. In addition, other properties of inscribed quadrilaterals and related problems are considered. В данной работе приведено простое доказательство одного результата из недавней работы Ю. Г. Никонорова и О. Ю. Никоноровой, первоначальное доказательство которого довольно громоздко и основано на исследовании полиномиальных идеалов и символьных вычислениях с использованием компьютера. Новое доказательство опирается на теорему Птолемея о вписанных четырехугольниках. В дополнение к этому рассмотрены другие свойства вписанных четырехугольников и смежные задачи. выпуклый многоугольник выпуклый четырехугольник вписанный четырехугольник теорема Птолемея экстремальные задачи целочисленный пятиугольник convex polygon convex quadrilateral extremal problems inscribed quadrilateral integer pentagon perimeter Ptolemy's theorem 1 Nikonorov, Yu. G. and Nikonorova, O. Yu.Some Extremal Problems for Polygons in the Euclidean Plane, Aequationes Mathematicae, 2024, vol. 98, no. 2, pp. 603-624. DOI: 10.1007/s00010-023-00991-w. Nikonorov, Yu. G. and Nikonorova, O. Yu.Some Extremal Problems for Polygons in the Euclidean Plane, Aequationes Mathematicae, 2024, vol. 98, no. 2, pp. 603-624. DOI: 10.1007/s00010-023-00991-w. 2 Muller, A. Auf Einem Kreis Liegende Punktmengen Ganzzahliger Entfernungen, Elem. Math., 1953, vol. 8, pp. 37-38. Muller, A. 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