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A note on an inequality involving the sides and medians in a triangle

This paper provides a proof for a previously unproven conjecture regarding an inequality involving the sides and medians of a triangle, extends the result to analogous inequalities involving altitudes and angle bisectors, and concludes by posing an open problem concerning Cevians.

Original authors: Peter Vassilev

Published 2026-03-19

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Peter Vassilev

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 you have a triangular tent. In geometry, this tent has three sides (let's call them the Sides) and three special lines running from each corner to the middle of the opposite side (called Medians).

For decades, mathematicians have been trying to solve a puzzle about these lines. A Bulgarian mathematician named Georgi Boychev noticed something strange in the 1980s: if you mix the lengths of the sides and the medians in a very specific, complicated recipe, the result always seems to be positive (greater than zero). He tried thousands of numbers to check it, and it always worked, but he couldn't find the "magic key" (the proof) to explain why it was true.

This paper is the story of how the author, Peter Vassilev, finally found that key.

The Big Puzzle: The "Magic Recipe"

The inequality in question looks like a scary math monster:
$(\sqrt{bc} - a)m_a + (\sqrt{ac} - b)m_b + (\sqrt{ab} - c)m_c \geq 0$

In plain English:
Imagine you take the square root of the product of two sides, subtract the third side, and multiply that by the median of that third side. Do this for all three corners and add them up. The author proves that no matter how you stretch or squish your triangle (as long as it doesn't collapse into a flat line), the total will never be negative. It's always a positive number.

How They Solved It: The "Weighted Scale" Analogy

To prove this, the author didn't just brute-force the math; he used a clever strategy involving balance scales.

The "Heavy" Corner: The author first proved a helper fact (Lemma 4). If your triangle has sides of different lengths ( $a < b < c$ ), the "weight" of the middle side ( $b$ ) combined with its median is the heaviest part of the equation. It's heavier than the smallest side's weight and heavier than the largest side's weight.
- Analogy: Think of the triangle as a seesaw. The middle side is the fulcrum that carries the most load.
The "Tipping" Strategy: The proof splits into two scenarios:
- Scenario A: The smallest side's weight is heavier than the largest side's.
- Scenario B: The largest side's weight is heavier.
In both cases, the author uses a famous mathematical rule called the AM-GM Inequality (Arithmetic Mean - Geometric Mean). Think of this rule as a law of averages: if you have three positive numbers, their average is always greater than or equal to their "geometric" average.

By replacing the "lighter" weights with the "heavier" middle weight, the author showed that even in the worst-case scenario, the "Magic Recipe" still tips the scale to the positive side.

The Twist: It Works for Other Lines Too!

The author didn't stop at medians. He asked: "Does this magic recipe work if we swap the medians for other special lines?"

Altitudes (The Drop): These are lines dropped straight down from a corner to the ground (perpendicular). Yes, the inequality holds.
Angle Bisectors (The Splitter): These are lines that cut the corner angle exactly in half. Yes, the inequality holds here too.

It's as if the triangle has a hidden "law of physics" that keeps this specific recipe positive, regardless of whether you use the median, the drop, or the splitter.

The Open Mystery: The "Cevian" Conundrum

The paper ends with a challenge for future mathematicians. There is a general family of lines called Cevians (which includes medians, altitudes, and bisectors).

The author found that for some weird, made-up Cevians (like "Splitters" that cut the perimeter in half), the inequality fails.

The Open Problem:
Can we find a specific type of Cevian line that follows the "rules of the road" (like the medians do—where the middle line is the heaviest) but still breaks the "Magic Recipe"?

The author suspects the answer is "yes," but hasn't found the specific example yet. It's like finding a car that has a working engine and wheels but still refuses to drive forward.

Summary

The Goal: Prove a long-standing guess about triangle sides and medians.
The Method: Used a "heaviest weight" strategy and the "Law of Averages" to show the math always balances out to a positive number.
The Discovery: This rule isn't just for medians; it works for altitudes and angle bisectors too.
The Future: Mathematicians are now challenged to find a "rogue" line that breaks this rule, even if it looks like it should follow it.

The paper is a tribute to the author's grandmothers, showing that the pursuit of mathematical truth is a journey that connects generations, much like the lines connecting the corners of a triangle.

1. Problem Statement

The paper addresses a long-standing conjecture regarding inequalities involving the sides ( $a, b, c$ ) and medians ( $m_a, m_b, m_c$ ) of a non-degenerate triangle. The primary focus is proving the validity of the following inequality, originally observed by Bulgarian mathematician Georgi Boychev in the 1980s but never formally proved:

$(\sqrt{bc} - a)m_a + (\sqrt{ac} - b)m_b + (\sqrt{ab} - c)m_c \geq 0 \quad \text{(Inequality 1)}$

The author also investigates a related inequality proposed by Kee-Wai Lau:
$(bc - a^2)m_a + (ac - b^2)m_b + (ab - c^2)m_c \geq 0 \quad \text{(Inequality 2)}$

Furthermore, the paper explores whether these inequalities hold when medians are replaced by other types of Cevians, specifically altitudes and internal angle bisectors, and poses an open problem regarding general Cevians.

2. Methodology

The proof strategy relies on a combination of algebraic manipulation, geometric properties of triangles, and the Arithmetic Mean-Geometric Mean (AM-GM) inequality. The methodology proceeds through the following steps:

Reduction to Specific Cases: The author first establishes that the inequality holds for isosceles triangles (Lemma 1). This reduces the general problem to the case of scalene triangles where $a < b < c$ .
Utilization of Known Inequalities: The proof heavily relies on a known system of inequalities relating medians and sides (Inequality 3 in the text):
$2am_a \leq cm_b + bm_c, \quad \text{etc.}$
This system is crucial for bounding terms in the main expression.
Monotonicity and Ordering: The author proves that for $a < b < c$ , the medians satisfy $m_a > m_b > m_c$ . Crucially, Lemma 4 establishes a specific ordering for the products of sides and medians:
$B = b m_b > \max(A = a m_a, C = c m_c)$
This ordering allows the author to manipulate the terms in the inequality by substituting the maximum value ( $B$ ) to lower-bound the expression.
Case Analysis and AM-GM: The proof splits into cases based on the relative magnitudes of $A$ and $C$ . By rearranging the original inequality into forms like:
$\left(\frac{\sqrt{c}m_a}{\sqrt{b}m_b} - 1\right)B + \dots \geq 0$
The author utilizes the fact that $B$ is the dominant term. By applying the AM-GM inequality (specifically that if the product of three positive numbers is 1, their sum is $\geq 3$ ), the author demonstrates that the sum of the coefficients is non-negative, thereby proving the inequality.
Extension to Other Cevians: Similar logic is applied to altitudes ( $h$ ) and angle bisectors ( $l$ ). For altitudes, the proof simplifies to standard algebraic inequalities. For angle bisectors, a similar lemma regarding the ordering of products ( $b l_b \geq \max(a l_a, c l_c)$ ) is established to replicate the median proof structure.

3. Key Contributions

Proof of Conjecture: The primary contribution is the rigorous proof of Inequality (1), a problem that had remained unresolved since the 1980s despite numerical verification.
Unification of Inequalities: The paper demonstrates that Inequality (1) and the related Inequality (2) are not isolated to medians but are valid for altitudes and internal angle bisectors as well.
Generalization to Linear Combinations: The author shows that the inequalities hold for any linear combination of medians, altitudes, and angle bisectors with non-negative coefficients (Corollary 2).
Identification of Necessary Conditions: The paper identifies that the validity of these inequalities depends on specific ordering properties of the Cevians (specifically $C_a \geq C_b \geq C_c$ and $b C_b \geq \max(a C_a, c C_c)$ ).

4. Results

Theorem 1: Confirms that $(\sqrt{bc} - a)m_a + (\sqrt{ac} - b)m_b + (\sqrt{ab} - c)m_c \geq 0$ holds for all non-degenerate triangles.
Extension Results: The analogous inequalities are proven true for:
- Altitudes ( $h_a, h_b, h_c$ ).
- Internal angle bisectors ( $l_a, l_b, l_c$ ).
- Any convex combination of these Cevians.
Counterexamples and Open Problems:
- The paper notes that these inequalities do not hold for arbitrary Cevians.
- Open Problem 1: The author poses a question regarding "Splitters" (Cevians passing through the Nagel point). While Splitters satisfy the necessary ordering conditions ( $C_a \geq C_b \geq C_c$ and $b C_b \geq \max(a C_a, c C_c)$ ), the author provides a counterexample (an obtuse triangle with sides 4, 16, 18) where these conditions are met, yet the inequalities (1) and (2) fail.
- This suggests that while the ordering conditions are necessary, they are not sufficient for the inequalities to hold.

5. Significance

Resolution of a Historical Problem: The paper resolves a conjecture that had persisted for decades, bridging a gap between numerical experimentation and formal proof in triangle geometry.
Methodological Insight: It highlights the power of combining geometric ordering properties (monotonicity of medians/bisectors) with algebraic inequalities (AM-GM) to solve complex geometric problems.
New Research Direction: By identifying that the inequalities fail for Splitters despite satisfying the ordering conditions, the paper opens a new avenue of research into the specific properties required for Cevians to satisfy such inequalities. It challenges mathematicians to find the precise necessary and sufficient conditions for general Cevians.
Educational Value: The paper serves as a resource for understanding the interplay between triangle centers, side lengths, and inequality theory, referencing standard texts on inequalities as further reading.