Some Geometric Properties of the Yff Points of a Triangle

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.

💡 Research Summary

The paper revisits the Yff points of a triangle, originally introduced by Peter Yff in 1963, and investigates new geometric properties of these points using modern computer‑assisted methods. The Yff points Y₁ and Y₂ are defined via six points D₁, D₂, E₁, E₂, F₁, F₂ placed on the sides of triangle ABC such that the distances AF₁ = BD₁ = CE₁ = AE₂ = BF₂ = CD₂ = u, where u is the real root of the cubic equation x³ = (a−x)(b−x)(c−x) with a, b, c the side lengths. The paper first recalls the known result that Y₁Y₂ is perpendicular to the line joining the first and third triangle centers X₁ (the incenter) and X₃ (the circumcenter).

The authors adopt a two‑stage computational approach. First, they generate random triangles and, for integer indices n, m ranging from 1 to 30, they numerically test whether various lines involving Y₁, Y₂, and the classical triangle centers Xₙ, Xₘ satisfy parallelism, orthogonality, equal length, or equal angle conditions. No new relations appear for arbitrary triangles beyond the known perpendicularity. Second, they focus on special families of triangles—right triangles, right triangles with a 30° angle, triangles with a 60° angle, “heptagonal” triangles whose angles are π/7, 2π/7, and 4π/7, and harmonic triangles (one side equal to the harmonic mean of the other two). For each family they discover additional concurrency, parallelism, or ratio properties involving Y₁, Y₂, and various X‑centers.

All results are proved symbolically using barycentric coordinates. The authors express each point in barycentric form, represent lines by the cross product of two point vectors, and formulate concurrency as the vanishing of a 3×3 determinant. The presence of the parameter u leads to high‑degree polynomial equations; the authors eliminate u by Gröbner‑basis computation (Mathematica’s Eliminate function), reducing the conditions to pure relations among a, b, c. For example, in a right triangle with the right angle at B, the concurrency of AY₂, CY₁, and X₈X₂₀ reduces to the factor a²−b²+c² = 0, which is precisely the Pythagorean condition. Similar eliminations yield conditions that match the law of cosines for 30° and 60° cases, and trigonometric identities for the heptagonal case (sin(π/7), sin(2π/7), sin(4π/7)).

The paper presents the following principal theorems:

• Theorem 2.1: In a right triangle (∠B = 90°), the lines AY₂, CY₁, and X₈X₂₀ are concurrent.
• Theorem 2.2: In a right triangle with ∠ACB = 30°, the lines AY₁, BY₂, and X₈X₂₁ are concurrent.
• Theorem 2.4: In a right triangle, BX₈ is perpendicular to Y₁Y₂, using the fact that BX₈ and X₁X₃ share the same point at infinity.
• Theorem 3.1: In any triangle with ∠C = 60°, AY₁, BY₂, and X₃X₈ are concurrent.
• Theorems 4.3–4.6: For a “heptagonal” triangle (angles π/7, 2π/7, 4π/7), various parallelisms and angle equalities hold among reflections of Y₁, the incenter X₁, the circumcenter X₃, and other X‑centers (X₂₁, X₂₈, etc.). The proofs rely on eliminating u and substituting the sine ratios that define the side lengths.
• Theorems 5.1–5.5: When Y₁Y₂ passes through a vertex, it is a median; when B Y₁ is a median, Y₁ divides it in the ratio 2b : 2a−b; analogous statements hold for Y₂. Moreover, a triangle is harmonic (one side equals the harmonic mean of the other two) if and only if Y₁Y₂ passes through a vertex.

Each theorem is accompanied by a “pure synthetic proof?” open question, highlighting that the current proofs are heavily computational. The authors also list three additional theorems (4.7–4.9) involving other triangle centers (X₅, X₂₁, X₂₈) whose proofs are omitted due to their computational intensity.

In conclusion, the paper demonstrates that computer‑assisted exploration can uncover non‑trivial geometric relationships involving the relatively obscure Yff points. By translating the problem into barycentric coordinates and leveraging symbolic elimination, the authors provide rigorous algebraic proofs for a variety of special triangle families. However, the reliance on massive symbolic computation leaves the classical synthetic geometry community with open challenges: to find elegant, coordinate‑free proofs for the presented results and to deepen the conceptual understanding of why Yff points interact so richly with classical triangle centers in these special configurations. The work thus opens a fertile avenue for future research at the intersection of computational algebra and synthetic Euclidean geometry.