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Some Geometric Properties of the Yff Points of a Triangle

This paper presents and proves several new geometric properties of the Yff points of triangles with various shapes, utilizing computer-assisted discovery to expand upon the limited knowledge established since their introduction in 1963.

Original authors: Stanley Rabinowitz, Francisco Javier García Capitán

Published 2026-02-05

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CC BY 4.0

Original authors: Stanley Rabinowitz, Francisco Javier García Capitán

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a triangle not just as a shape on a piece of paper, but as a tiny, bustling city with three corners (A, B, and C) and three streets connecting them. For decades, mathematicians have been mapping out special "landmarks" inside these cities. One such pair of landmarks, discovered in 1963 by a man named Peter Yff, are called the Yff Points (let's call them Y1 and Y2).

Think of Y1 and Y2 as two very precise GPS coordinates that appear inside every triangle. They are created by a specific, somewhat mysterious rule involving the lengths of the streets. If you draw lines from the corners to these points, they meet in a very specific way.

For a long time, very little was known about these two points. But in this paper, the authors (Stanley Rabinowitz and Francisco Javier García Capitán) decided to put on their "digital detective hats." They used a powerful computer program (Mathematica) to simulate thousands of different triangles—some skinny, some fat, some right-angled, some with sides that follow a specific pattern—to see what new secrets Y1 and Y2 might be hiding.

Here is what they found, translated into everyday language:

1. The Computer as a Crystal Ball

The authors didn't just guess; they ran a massive simulation. They threw random triangles at the computer, added the Yff points, and asked the computer: "Do these lines cross at the same spot? Are these lines parallel? Are these angles equal?"

The Analogy: Imagine throwing thousands of different-shaped kites into the wind and asking a super-fast camera, "Do the strings of these kites always cross at the same point?"
The Result: When the computer said "Yes, they look equal to 10 decimal places," the authors then switched to "math mode" to prove it was true for all triangles, not just the ones the computer checked.

2. Special Shapes, Special Secrets

The paper reveals that the Yff points behave differently depending on the "personality" of the triangle:

Right Triangles (The 90-degree Corner): If your triangle has a perfect 90-degree corner, the Yff points form a secret handshake with other famous landmarks (like the Nagel Point and the de Longchamps Point). Specifically, certain lines connecting them meet at a single point, or stand perfectly perpendicular (at right angles) to each other.
The "Heptagonal" Triangle (The 7-sided Mystery): There is a rare triangle where the angles are related to the number 7 (specifically $\pi/7$ , $2\pi/7$ , and $4\pi/7$ ). In this specific shape, the Yff points do something magical: if you reflect one of them across a side, it becomes perfectly parallel to a line connecting two other famous points. It's like a geometric magic trick that only works for this specific shape.
Harmonic Triangles (The Balanced Scale): A "harmonic triangle" is one where one side is the "harmonic mean" of the other two (a specific mathematical average). The paper proves that in these triangles, the line connecting Y1 and Y2 doesn't just float randomly; it acts like a median. It goes straight through a corner and hits the exact middle of the opposite side. It's as if the Yff points are the "balance beam" of the triangle.
Arithmetic Progression Triangles (The Step Ladder): If the sides of the triangle are numbers like 3, 4, and 5 (where each step is the same size), the line connecting the two Yff points runs perfectly parallel to a line drawn from a corner to the "incenter" (the center of the circle that fits inside the triangle).

3. The Area of the "Yff Triangle"

The authors also looked at the tiny triangle formed by connecting the points where the lines from the corners to Y1 hit the opposite sides. They found a beautiful formula for the area of this tiny inner triangle.

The Metaphor: Imagine the Yff points are the seeds of a flower. The paper calculates exactly how big the petals of this flower will be based on the size of the original triangle. The formula is surprisingly clean: the area depends on a specific number ( $u$ ) cubed, divided by the size of the triangle's "circumcircle" (the circle that wraps around the whole triangle).

4. The "Open Questions" (The Unsolved Mysteries)

Even though the computer proved these facts, the authors admit that the proofs feel a bit like "black box" magic. The computer did millions of calculations to show the numbers matched, but it didn't give a simple, visual reason why it happens.

The Challenge: The paper asks: "Can we find a purely visual, logical proof for these facts without using a computer?" It's like asking, "We know the bridge holds, but can we explain the physics of the arch without doing the math?"

Summary

In short, this paper is a treasure map. It uses a computer to find hidden geometric patterns involving two special points (Y1 and Y2) inside triangles. It shows that when triangles have specific shapes (like right angles, specific angles related to the number 7, or balanced sides), these points line up, cross, or run parallel in surprising and elegant ways. While the computer confirmed the patterns, the authors are still looking for the simple, human-readable story behind why these geometric relationships exist.

Technical Summary: Some Geometric Properties of the Yff Points of a Triangle

Problem Statement
The Yff points ( $Y_1$ and $Y_2$ ) of a triangle, introduced by Peter Yff in 1963, are defined by the concurrency of cevians constructed from points on the triangle's sides such that specific segments equal a real root $u$ of the cubic equation $x^3 = (a-x)(b-x)(c-x)$ . Despite their definition, the literature contains very few facts about these points beyond Yff's original findings, which established their existence, their status as isotomic conjugates, and the orthogonality of the line connecting them to the line connecting the incenter ( $X_1$ ) and circumcenter ( $X_3$ ). The paper addresses the lack of known geometric properties regarding Yff points in triangles of specific shapes and configurations.

Methodology
The authors employ a "computer-discovered mathematics" approach, utilizing the software Mathematica to generate and verify geometric conjectures. The process involves two distinct phases:

Numerical Exploration: The authors generate random numerical triangles and specific classes of triangles (e.g., right, heptagonal, arithmetic progression). They numerically check for geometric coincidences (parallelism, concurrency, perpendicularity, equal angles) involving the Yff points and various known triangle centers ( $X_n$ ) for $n, m \in [1, 30]$ .
Symbolic Verification: Once a property is identified numerically (valid to 10 decimal places), the authors switch to exact symbolic computation. They utilize barycentric coordinates, vector cross products for line intersections, and elimination theory (specifically Gröbner bases and the Eliminate function) to formally prove the results. This involves substituting the algebraic definition of $u$ (or eliminating $u$ via the defining cubic equation) to verify that geometric conditions hold identically for the specific triangle classes.

Key Contributions and Results
The paper presents a series of theorems establishing new geometric properties of Yff points across various triangle classifications:

General Properties: The paper re-confirms that $Y_1$ and $Y_2$ are isotomic conjugates and that $Y_1Y_2 \perp X_1X_3$ . It poses the open question of whether a purely synthetic proof exists for the latter.
Right Triangles:
- In a right triangle with the right angle at $B$ , the lines $AY_2$ , $CY_1$ , and $X_8X_{20}$ are concurrent.
- In a right triangle with $\angle B = 90^\circ$ and $\angle C = 30^\circ$ , the lines $AY_1$ , $BY_2$ , and $X_8X_{21}$ are concurrent.
- In a right triangle with the right angle at $B$ , the line $BX_8$ is perpendicular to $Y_1Y_2$ .
60° Triangles: In a triangle with $\angle C = 60^\circ$ , the lines $AY_1$ , $BY_2$ , and $X_3X_8$ are concurrent.
Heptagonal Triangles: (Triangles with angles $\pi/7, 2\pi/7, 4\pi/7$ $π /7, 2 π /7, 4 π /7$ )
- If $Y'_1$ is the reflection of $Y_1$ across side $BC$, then $X_1Y_2 \parallel Y'_1C$ .
- Specific angle relationships are established, such as $\angle CX_1Y_2 + \angle X_1CY_1 = \pi/7$ and $\angle CX_1Y_2 = \angle Y_1CX_3$ .
- Several other angle equalities involving centers $X_5, X_{21}, X_{28}$ are noted, though proofs are omitted due to computational complexity.
Harmonic Triangles: A triangle is "harmonic" (one side is the harmonic mean of the other two) if and only if the line $Y_1Y_2$ passes through a vertex, the incenter ( $X_2$ ), or bisects a side. In such cases, $Y_1Y_2$ coincides with a median.
Arithmetic Progression (AP) Triangles: In a triangle where sides are in arithmetic progression ( $2a = b+c$ ), the line $AX_1$ is parallel to $Y_1Y_2$ .
Double-Angle Triangles: In a triangle where $\angle A = 2\angle B$ , the lines $BY_2$ , $CX_2$ , and the reflection of $AY_2$ across $BC$ are concurrent.
Isosceles Triangles: The product of the areas of triangles formed by $Y$ and the sides ( $K_b K_c$ ) equals the square of the third area ( $K_a^2$ ) if and only if the triangle is isosceles with $AB=AC$.
Area of Cevian Triangles: The authors provide a purely geometric proof for Yff's formula regarding the area of the cevian triangle of $Y_1$ (and $Y_2$ ), showing it equals $u^3 / (2R)$ , where $R$ is the circumradius.
Special Points: The paper identifies the midpoint of $Y_1Y_2$ and the center of the conic passing through $A, B, C, Y_1, Y_2$ as new triangle centers not currently cataloged in the Encyclopedia of Triangle Centers (ETC).

Significance and Claims
The authors modestly frame the paper's contribution as the discovery of "some geometrical properties" that were previously unknown. The significance lies in expanding the limited body of knowledge regarding Yff points, which had remained largely static since 1963. The paper explicitly acknowledges that the initial findings were numerical and that the formal proofs rely heavily on computer algebra systems, resulting in "enormous" computations that lack intuitive synthetic elegance. Consequently, the paper concludes by posing open questions regarding the existence of "purely synthetic geometric proofs" for the presented theorems, particularly for the right triangle and heptagonal triangle results. The work serves as a catalog of verified properties for specific triangle classes, inviting further theoretical investigation into the underlying geometry.

1. The Computer as a Crystal Ball

2. Special Shapes, Special Secrets

3. The Area of the "Yff Triangle"

4. The "Open Questions" (The Unsolved Mysteries)

Summary

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