One application of Ptolemy's theorem

Yu. G. Nikonorov^a, D. N. Oskorbin<sup>b

a</sup> Southern Mathematical Institute of VSC RAS, 53 Vatutin St., Vladikavkaz 362025, Russia
^b Moscow Institute of Physics and Technology, 9 Institutsky Per., Dolgoprudny 141701, Russia

Abstract: 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.

Key words: convex polygon, convex quadrilateral, extremal problems, inscribed quadrilateral, integer pentagon, perimeter, Ptolemy's theorem.

UDC: 514.14

MSC: 52A10, 52A40, 52B60

Received: 25.03.2025

Language: English

DOI: 10.46698/o1944-0990-7792-e