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 with the reference triangle, and in each case, the corresponding three perspectors are collinear. In this note, we provide proof of his assertions. Furthermore, as an analogue of Lemoine’s problem, we formulated and answered the question of how to recover the reference triangle given a Kiepert hyperbola, one of the two Fermat points and one vertex of the reference triangle.
This paper establishes rigorous proofs for a set of remarkable geometric properties previously conjectured by Paul Yiu in 2014 concerning equilateral triangles associated with the Kiepert hyperbola. The work validates and extends a significant construction in modern triangle geometry, demonstrating a profound interplay between classical Euclidean concepts and advanced projective techniques.
At its core, the paper focuses on the Kiepert hyperbola, a unique conic defined for any given reference triangle ABC. This hyperbola is known to pass through several notable triangle centers, including the vertices A, B, C, the first and second Fermat points (F₁ and F₂), and the centroid. Yiu’s original assertion was that two specific equilateral triangles can be inscribed within this Kiepert hyperbola. These triangles, PQR and P’Q’R’, are generated by intersecting circles centered at the Fermat points (F₂ and F₁) with the Kiepert hyperbola itself. The paper proves two main properties of these triangles:
1. Equilaterality: PQR and P’Q’R’ are indeed equilateral.
2. Triple Perspectivity and Collinearity: Each of these equilateral triangles (PQR and, with a specific orientation, P’R’Q’) is triply perspective to the original reference triangle ABC. This means that for each pair of triangles, there are three distinct choices of corresponding vertices (e.g., (A,P,B,Q,C,R), (A,Q,B,R,C,P), and (A,R,B,P,C,Q)) such that the lines joining corresponding vertices (e.g., AP, BQ, CR) are concurrent. Remarkably, the paper proves that the three points of concurrency (perspectors) arising from each triple perspectivity are collinear.
The significance of this work lies in providing the mathematical bedrock for Yiu’s elegant discoveries, moving them from assertion to proven fact. It enriches the field of triangle geometry by adding new, non-trivial constructions and relationships involving fundamental triangle centers and conics. The paper also highlights the aesthetic appeal of geometric theorems where multiple, seemingly independent properties (equilaterality, triple perspectivity, collinearity) converge.
Key innovations in this paper include:
* Rigorous Proofs: The central innovation is providing definitive, detailed proofs for Yiu’s assertions, which had previously lacked formal justification.
* Analogue of Lemoine’s Problem: The paper formulates and solves an inverse problem akin to Lemoine’s classic construction problem. Given a Kiepert hyperbola, one of the two Fermat points, and one vertex of the reference triangle, the paper demonstrates how to reconstruct the other two vertices of the original triangle. This adds a valuable constructive aspect to Yiu’s results.
* Methodological Synthesis: The paper effectively employs a blend of geometric techniques:
* Synthetic Geometry: Used for initial proofs, drawing on established theorems like Feuerbach’s Theorem.
* Analytic (Coordinate) Geometry: Extensively used in the core proofs (Theorem 2) involving rational parameterization of circles and a Kiepert hyperbola, often relying on computer algebra systems (SAGE) for complex algebraic manipulations. This showcases the power of computational methods in modern geometry research.
* Projective Geometry: Utilized to provide a more general framework for results on triply perspective triangles inscribed in a conic, establishing broader implications beyond the specific context of the Kiepert hyperbola.
The main prior ingredients upon which this research builds are:
* Yiu’s Original Construction (2014): The assertions and the geometric setup that forms the basis of the paper.
* Classical Triangle Geometry: Fundamental concepts such as:
* Fermat Points (Isogonic Centers): Their definitions and properties are crucial for defining the associated circles and the Kiepert hyperbola.
* Kiepert Hyperbola: Knowledge of its definition (passing through vertices, Fermat points, centroid), its nature as a rectangular hyperbola, and its property that the orthocenter of an inscribed triangle lies on it are essential. Its center being the midpoint of the Fermat points is also key.
* Nine Point Circle (Feuerbach Circle): Utilized in the proof of equilaterality, specifically its relationship with rectangular hyperbolas and their centers (Feuerbach’s Theorem).
* Projective Geometry Concepts:
* Pascal’s Theorem: Repeatedly applied to establish collinearity relationships, particularly in the context of perspective triangles and Hessian lines.
* Perspective Triangles: The theory of triangles being perspective from a point (or axis), including the concept of doubly and triply perspective triangles.
* Projective Collineations: Employed to simplify general proofs by mapping complex conics to simpler forms (like circles) and then transforming back.
* Analytic Geometry: The ability to represent geometric objects (points, lines, conics) with coordinates and perform algebraic computations, which is fundamental to the detailed proofs in Section 3.