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Some Geometric Properties of the Yff Points of a Triangle

Stanley Rabinowitz, Francisco Javier García Capitán

TL;DR

The article advances the study of Yff points by combining computer-assisted discovery and symbolic verification to derive new geometric properties across diverse triangle shapes, including right, $60^ extot}$, heptagonal, harmonic, AP, double-angle, and isosceles cases. It establishes new concurrency and parallelism results, explicit barycentric and simple coordinates for central points, and an exact area formula for the Yff cevian triangle $[DEF]=\frac{u^3}{2R}$, enriching connections to classical centers. The work also identifies new candidate centers, such as $Y_m$ and $Y_c$, and highlights the role of computational tools (e.g., Gröbner bases, elimination) in generating and validating geometric conjectures. Overall, the paper expands the geometric panorama surrounding Yff points, linking them to broader triangle center theory and demonstrating the effectiveness of computer-assisted synthetic geometry.

Abstract

The Yff points of a triangle were introduced by Peter Yff in 1963. Since then, very few new facts have been discovered about these points. We present some geometrical properties of the Yff points of various shaped triangles which were discovered and proved by computer.

Some Geometric Properties of the Yff Points of a Triangle

TL;DR

The article advances the study of Yff points by combining computer-assisted discovery and symbolic verification to derive new geometric properties across diverse triangle shapes, including right, , heptagonal, harmonic, AP, double-angle, and isosceles cases. It establishes new concurrency and parallelism results, explicit barycentric and simple coordinates for central points, and an exact area formula for the Yff cevian triangle , enriching connections to classical centers. The work also identifies new candidate centers, such as and , and highlights the role of computational tools (e.g., Gröbner bases, elimination) in generating and validating geometric conjectures. Overall, the paper expands the geometric panorama surrounding Yff points, linking them to broader triangle center theory and demonstrating the effectiveness of computer-assisted synthetic geometry.

Abstract

The Yff points of a triangle were introduced by Peter Yff in 1963. Since then, very few new facts have been discovered about these points. We present some geometrical properties of the Yff points of various shaped triangles which were discovered and proved by computer.
Paper Structure (12 sections, 34 theorems, 48 equations, 26 figures)

This paper contains 12 sections, 34 theorems, 48 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Right Triangles
  3. $60^\circ$ Triangles
  4. Heptagonal Triangles
  5. Harmonic Triangles
  6. AP Triangles
  7. Double-Angle Triangles
  8. Isosceles Triangles
  9. Isosceles Right Triangles
  10. Special Triangles
  11. Cevian Triangles
  12. Special Points

Key Result

Theorem 1.1

Let $D_1$, $D_2$, $E_1$, $E_2$, $F_1$, $F_2$ be points on the sides of triangle $ABC$ (as shown in Figure fig:YffPoints) such that $AE_1=BF_1=CD_1=AF_2=BD_2=CE_2=u$ where $u$ is the real root of the equation $x^3=(a-x)(b-x)(c-x)$. Then $AD_1$, $BE_1$, $CF_1$ meet in a point $Y_1$ and $AD_2$, $BE_2$,

Figures (26)

  • Figure 1: Yff Points
  • Figure 2: red lines are perpendicular
  • Figure 3: dashed lines are concurrent
  • Figure 4: dashed lines are concurrent
  • Figure 5: blue lines are perpendicular
  • ...and 21 more figures

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • Theorem 2.4
  • proof
  • Theorem 3.1
  • ...and 49 more