The Triangle

TL;DR

This paper explores properties of orthocentric quadrangles derived from triangles, focusing on a special iterative process involving reflections and circumcircles, revealing new geometric sequences and their characteristics.

Contribution

It introduces a novel iterative geometric process involving reflections and circumcircles in orthocentric quadrangles, expanding understanding of their properties.

Findings

Identification of a trisequence with unique properties

Establishment of collinearity involving reflections and orthocentres

Insights into iterative geometric constructions in orthocentric systems

Abstract

If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle. Steiner's theorem states that the reflexions of a point on a circumcircle in each of the three edges of the corresponding triangle are collinear and collinear with its orthontre. This line intersects the circumcircles in new points to which the theorem may be applied. Iteration of this process with the triangle and the points rational leads to a "trisequence" whose properties merit study.