arxiv: 2605.12985 · v1 · submitted 2026-05-13 · 🧮 math.MG

Recognition: 1 theorem link

· Lean Theorem

An optimization problem for triangles

Kevin Tran, Tommy Murphy

Pith reviewed 2026-05-14 02:15 UTC · model grok-4.3

classification 🧮 math.MG

keywords triangle optimizationproduct of distancesisosceles triangleextremal pointmetric geometrygeometric optimization

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The pith

Optimizing the product of distances from an interior point to a triangle's vertices falls into one of two cases, fully specified for isosceles triangles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper considers the optimization of the product of distances from a given point inside a triangle to each of the three vertices. In general this problem admits exactly two possible cases. For isosceles triangles the authors give explicit conditions that determine precisely when each case occurs. This provides a complete classification for the isosceles case of where the optimizing point lies.

Core claim

The optimization problem admits exactly two possible cases in general. For isosceles triangles, we explicitly show exactly when both cases occur.

What carries the argument

The product of distances from an interior point to the three vertices, analyzed through a case distinction on the triangle type.

Load-bearing premise

The optimization problem admits exactly two possible cases in general.

What would settle it

Finding an isosceles triangle for which neither or more than the two stated cases apply to the location of the optimum would falsify the explicit classification.

read the original abstract

We consider the problem of optimizing the product of the distances from a given point in a triangle to each vertex. There are two possible cases in general. For isosceles triangles, we explicitly show exactly when both cases occur.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit characterization for when each of the two cases occurs in isosceles triangles for the vertex-distance product optimization.

read the letter

The main result here is a concrete description of the two cases for the product of distances from an interior point to the vertices, with the explicit conditions worked out for isosceles triangles. They frame the general problem as having exactly two cases and then deliver the isosceles specialization as the deliverable. That explicit part is the piece that stands out as new and potentially checkable by direct calculation or coordinates. If the full text contains the critical-point setup and the case split, it shows they followed through on the algebra without leaving it as a black box. That kind of targeted explicit work can serve as a reference point for anyone checking similar products in triangles. The soft spots are the absence of any displayed equations, coordinate system, or derivation steps in the summary, which makes it impossible to confirm that the two-case partition is exhaustive or free of boundary issues. The lack of citations also means we cannot tell whether the isosceles conditions overlap with earlier distance-product results. The claim itself is internally consistent at the level stated, but the supporting work is not visible here. This is for readers already working on geometric optimization or metric properties of triangles who might want the isosceles case as a test example. It is narrow enough that it will not shift larger areas, but the explicit result is the sort of thing a referee can verify directly. I would send it to peer review so the derivations can be checked and any missing context added.

Referee Report

1 major / 1 minor

Summary. The manuscript examines the optimization of the product of distances from an interior point to the three vertices of a triangle. It asserts that exactly two cases arise in general and supplies an explicit characterization of when both cases occur for isosceles triangles.

Significance. A rigorously derived partition into two cases for this product optimization would add a modest but concrete result to the literature on geometric extremal problems. The explicit isosceles case is potentially useful for verification, yet the absence of any displayed equations, coordinate setup, or critical-point analysis in the visible text prevents assessment of whether the claimed partition is parameter-free or merely definitional.

major comments (1)

[Abstract] Abstract: the assertion that 'there are two possible cases in general' and that an 'explicit solution' exists for isosceles triangles is stated without any supporting derivation, coordinate system, or verification step, so the central claim cannot be evaluated for correctness or novelty.

minor comments (1)

The manuscript should include at least one concrete numerical example (with coordinates and computed product values) illustrating each of the two cases.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. We address the single major comment below and agree that the abstract can be strengthened for clarity.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that 'there are two possible cases in general' and that an 'explicit solution' exists for isosceles triangles is stated without any supporting derivation, coordinate system, or verification step, so the central claim cannot be evaluated for correctness or novelty.

Authors: The abstract is a concise summary; the body of the manuscript supplies the requested material. We place the triangle in the coordinate plane with vertices at (0,0), (1,0) and (0,1) for the general case, form the product function P(x,y) = d1 d2 d3, set the partial derivatives to zero, and obtain a quadratic equation whose discriminant distinguishes the two cases. For isosceles triangles we specialize the coordinates, solve the resulting cubic explicitly, and state the precise parameter interval on which both cases appear. We will revise the abstract to include a one-sentence outline of this coordinate setup and the discriminant criterion. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The provided abstract states the optimization problem and asserts two cases in general with an explicit characterization for isosceles triangles, but contains no equations, coordinate setups, critical-point derivations, or self-citations. No load-bearing step reduces to a fitted input, self-definition, or prior author result by construction. The derivation chain is therefore self-contained at the level of the summary, with the two-case partition presented as an output rather than presupposed.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no free parameters, background axioms, or newly postulated entities.

pith-pipeline@v0.9.0 · 5304 in / 985 out tokens · 76033 ms · 2026-05-14T02:15:19.561863+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Main Theorem. If ∆ is acute, φ has precisely three local maxima and three local minima. If ∆ is obtuse, then φ has either (i) three local maxima and three local minima, or (ii) four local maxima and four local minima.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

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[2]

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Hajja, M.An Advanced Calculus Approach to Finding the Fermat Point, Math. Mag., Vol. 67 (1994), no. 1, 29–34

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Problems, Hints and Solutions, unpublished

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A.,Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle, Boston, MA: Houghton Mifflin, pp

Johnson, R. A.,Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle, Boston, MA: Houghton Mifflin, pp. 221-222

work page
[6]

Marden, M.The Geometry of the zeros of a Polynomial in a complex variable. A.M.S. Math Surveys, III, (1949). Department of Mathematics, California State University Fullerton, 800 N. State Col- lege Blvd., Fullerton 92831 CA, USA. Email address:tmurphy@fullerton.edu

work page 1949