Sections of triangles and circles
type: Item
instance of: scholarly article
title: Sections of triangles and circles
zbMATH Open document ID: 1504.51013
DOI: 10.4171/EM/454
author: Peter Thurnheer
published in: Elemente der Mathematik
publication date: 2022-12-14T00:00:00Z
review text: Consider any triangle \(ABC\). Let \(G_A\) be any point on the median line \(s_A\) through \(A\) and midpoint \(M\) of \(BC\), excepting \(A\) and \(M\). Denoting \(E\) (resp. \(F\)) the intersection point of the lines connecting \(G_A\) with \(B\) (resp. \(C\)) and \(A\) with \(C\) (resp. \(B\)), call \(A\)-triangles the four triangles \(ABM\), \(AMC\), \(ABE\) and \(ACF\). Then the following three properties are shown. The centers of the circumscribed circle of each of the four \(A\)-triangles lie on a circle, which we call an \(A\)-circle. The locus of all centers of \(A\)-circles when \(G_A\) moves along \(s_A\) is a straight line parallel to \(s_A\), which we call the \(A\)-line. The \(A\)-line, \(B\)-line and \(C\)-line are concurrent.
reviewed by: Frank Plastria
Mathematics Subject Classification ID: 51M04
Mathematics Subject Classification ID: 51M15
zbMATH DE Number: 7633071
zbMATH Keywords: median
zbMATH Keywords: circumcircle
zbMATH Keywords: locus
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/em/454
OpenAlex ID: W3168293216
cites work: Q5292719
cites work: Q4709907
description: scientific article; zbMATH DE number 7633071