On Yiu's Equilateral Triangles Associated with a Kiepert Hyperbola

TL;DR

This paper proves Yiu's assertions about equilateral triangles inscribed in a Kiepert hyperbola, showing their perspectivity properties and addressing a problem related to reconstructing the reference triangle from given elements.

Contribution

It provides a proof of Yiu's geometric assertions and formulates and solves an analogue of Lemoine's problem for the Kiepert hyperbola.

Findings

Confirmed triply perspective property of equilateral triangles

Established collinearity of perspectors

Solved the reconstruction problem of the reference triangle

Abstract

In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective to the reference triangle, and in each case, the corresponding three perspectors are collinear. In this note, we give a proof of his assertions. Furthermore as an analogue of Lemoine's problem, we formulated and answered the question about how to recover the reference triangle given a Kiepert hyperbola, one of the two Fermat points and one vertex of the reference triangle.