An analogue of the Pythagorean theorem with regular \(n\)-gons instead of squares
type: Item
instance of: scholarly article
title: An analogue of the Pythagorean theorem with regular \(n\)-gons instead of squares
zbMATH Open document ID: 0877.51024
author: Darko Veljan
published in: Elemente der Mathematik
publication date: 1997-07-17T00:00:00Z
full work available at URL: https://eudml.org/doc/141599
review text: A triangle \(ABC\) with angle \(\gamma=2 \pi/n\) at \(C\) is said to be a fraction \(n\)-triangle, and one with \(\gamma= \pi-2 \pi/n\) is called a complementary fraction \(n\)-triangle. Furthermore, the side \(c\) opposite to \(C\) is the hypotenuse of \(ABC\), and the side \(c'= A'B\) of \(A'BC\) (where \(A'\) is obtained by reflecting \(A\) at \(C\), such that \(c'\) satisfies the equality \((c')^2= 2(a^2+ b^2)-c^2)\) is the corresponding complementary hypotenuse. The other two sides are called legs. The author proves the following Pythagorean theorem with regular \(n\)-gons: Let \(ABC\) be a fraction \(n\)-triangle (resp. comlementary fraction \(n\)-triangle). Then the area of a regular \(n\)-gon over the sum of the legs equals the sum of \(n\) areas of the triangle and the area of a regular \(n\)-gon over the complementary hypotenuse (resp. hypotenuse).
Mathematics Subject Classification ID: 51N05
Mathematics Subject Classification ID: 51M04
zbMATH DE Number: 1036783
zbMATH Keywords: affine regular polygons
zbMATH Keywords: cosine law
zbMATH Keywords: regular \(n\)-gons
zbMATH Keywords: Pythagorean theorem
reviewed by: Horst Martini
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1036783