The Deltoid Curve and Triangle Transformations

TL;DR

This paper explores the geometric properties of the deltoid curve related to triangle transformations, focusing on orthocenters, complex plane representations, and how raising vertices to powers affects the resulting curves and mappings.

Contribution

It introduces a novel analysis of triangle transformations involving power operations on vertices and their effects on orthocenters and associated curves, including the trifolium.

Findings

Points mapped to the deltoid lie on specific curves.

Varying the power parameter produces a family of curves, including the trifolium.

Points mapped to the origin are intersections of tangents to the deltoid.

Abstract

Deltoid curves appear as consequences of certain procedures in triangle geometry. The best known of these is the construction based on Simson lines, described by Steiner. This is carefully related, in this article, to a less known construction. The standard deltoid in the complex plane and its tangent lines are principle objects of study in this report. It is known that each point in the interior of this curve is the orthocenter of a triangle with distinct vertices on the unit circle, whose product is one. (If instead the point is on the deltoid, then at least two of the vertices coalesce, resulting in a degenerate triangle.) When the vertices are all raised to some specified integer power, a new (possibly degenerate) triangle results. By varying the triangle, one may thus consider the map taking the original triangle's orthocenter to the resulting triangle's orthocenter. Such maps are the other principle objects of study here. The points that are mapped to the deltoid lie on easily described curves. By varying the power involved in the map, a pleasing family of curves results, which includes a trifolium curve. The points that are mapped instead to the origin are described as the points of intersection of certain tangents to the deltoid.