arxiv: 2604.03054 · v1 · submitted 2026-04-03 · 🧮 math.MG

Recognition: no theorem link

A New Lemoine-Type Circle

Mi{\l}osz P{\l}atek

Pith reviewed 2026-05-13 18:33 UTC · model grok-4.3

classification 🧮 math.MG

keywords Lemoine circletriangle geometrycocyclicity criterionsix-point configurationTucker circlesBui circle

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The pith

A new Lemoine-type circle arises from six points on a triangle that meet a cocyclicity criterion.

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

The paper defines a fresh Lemoine-type circle through a six-point setup on the sides of a triangle. The points are required to lie on a single circle under a specific cocyclicity condition. The authors prove that this condition produces the circle, supply a converse result, compare the new object to earlier Lemoine circles including the one by Bui, and establish that it lies outside the Tucker family. A reader would care because the construction adds one more explicit circle to the list of notable loci tied to the Lemoine point in ordinary triangle geometry.

Core claim

The central claim is that a six-point configuration on the sides of a triangle, when the points satisfy the given cocyclicity criterion, determines a new Lemoine-type circle. The paper proves existence, states and proves a converse theorem, positions the circle relative to known examples such as Bui's Lemoine circle, and shows that the new circle does not belong to the Tucker family.

What carries the argument

The six-point configuration on the triangle sides together with its cocyclicity criterion, which together locate the new Lemoine-type circle.

If this is right

The cocyclicity criterion forces the six points to determine a single circle of Lemoine type.
The new circle is distinct from every member of the Tucker family.
A converse statement recovers the configuration from the circle.
The new circle shares some but not all properties with Bui's earlier Lemoine circle.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Multiple independent Lemoine-type circles may exist, suggesting a larger parametric family of such circles.
The six-point construction could be tested numerically on specific triangles to locate additional intersection properties.
Similar cocyclicity conditions might produce further circles attached to other classical triangle points.

Load-bearing premise

The six points satisfy the stated cocyclicity criterion that is used to define the new circle.

What would settle it

A concrete triangle together with an explicit choice of six points that obey all stated side conditions yet fail to lie on a common circle would falsify the general existence result.

Figures

Figures reproduced from arXiv: 2604.03054 by Mi{\l}osz P{\l}atek.

**Figure 1.** Figure 1: First Lemoine Circle Proposition 3.2. (Second Lemoine Circle) Let the lines through point L, which are antiparallel to the lines BC, CA, and AB, intersect lines CA, AB, and BC in six points. These six points lie on a single circle, called the Second Lemoine Circle of triangle ABC, and its center is point L. These two circles gave rise to the family of Lemoine circles. Their distinguishing feature is the f… view at source ↗

**Figure 2.** Figure 2: Second Lemoine Circle A B C L Cb Ca Ba Bc Ac Ab M O [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Third Lemoine Circle intersects lines AB and AC at points Ab and Ac, respectively. The circle ω2, passing through points B and L and tangent to ω at B, intersects lines BC and BA at points Bc and Ba, respectively. The circle ω3, passing through points C and L and tangent to ω at C, intersects sides CA and CB at points Ca and Cb, respectively. Then, the points Ab, Ac, Ba, Bc, Ca, and Cb lie on [PITH_FULL_I… view at source ↗

**Figure 4.** Figure 4: Q.T.Bui Circle Remark 3.5. Once again, for these circles, the only initial point for which these points lie on a single circle is point L, and in that case, the centers of those circles also lie on the Brocard axis. 4. New Lemoine-type Circle Here, we present the main result: a new Lemoine-type Circle, which fits nicely with the one discovered by Q.T. Bui in 2006. A unique property of this circle is that i… view at source ↗

**Figure 5.** Figure 5: Theorem 4.1. Proof. We will prove the following lemma: Lemma 4.2. Point L is the Lemoine point in triangle A′B′C ′ . Proof. Let us note that it is sufficient to prove that L lies on the symmedian from vertex A′ in triangle A′B′C ′ (the proof that it lies on the other symmedians will be analogous). Since the points A, L, A′ are collinear, it is enough to show that the quadrilateral AC′A′B′ is harmonic, or … view at source ↗

**Figure 6.** Figure 6: Points Ab, Ac, Ba, Bc, Ca, Cb are concyclic. We will now show that point M lies on LO and satisfies LM = 3 4 · LO [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: LM4 = M′O [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

**Figure 8.** Figure 8: Proof of Theorem 5.1. Theorem 5.1. Let P be any point inside triangle ABC with a circumcircle ω. Let A′ , B′ , C′ denote the intersections of the lines AP, BP, CP with the circle ω (distinct from A, B, C). The circle ω1 passing through points A′ and P, tangent to ω at point A′ , intersects side BC at points Ab and Ac. The circle ω2 passing through points B′ and P, tangent to ω at point B′ , intersects [PI… view at source ↗

**Figure 9.** Figure 9: Caption [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

read the original abstract

This paper presents a new Lemoine-type circle defined by a six-point configuration satisfying a cocyclicity criterion. We prove the result, establish a converse theorem, and relate the new circle to previously known Lemoine circles, in particular the one introduced by Q.T. Bui. We show that the new circle does not belong to the family of Tucker circles.

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

This paper defines a new six-point Lemoine-type circle via a cocyclicity criterion, proves the claim plus a converse, and shows it lies outside the Tucker family while linking to Bui's circle.

read the letter

The main contribution here is a concrete new circle in the Lemoine family, built from a six-point configuration that meets an explicit cocyclicity test. The authors prove the points are concyclic under that test, give the converse, and separate the construction from Tucker circles while tying it to Bui's earlier example. That separation and the converse are the parts that feel useful; they give the result a clear identity rather than just another named point set. The work sits squarely in classical triangle geometry and uses standard relations, so the argument looks internally consistent on the terms stated. No free parameters or circular definitions appear, and the claim is presented as a direct geometric fact rather than a fitted observation. The only soft spot is that the abstract gives no sample derivation or coordinate check, which leaves the cocyclicity step as a black box until the full write-up is read; if the paper supplies the usual angle-chasing or trigonometric identity, that gap closes. For readers already working on Lemoine or Tucker configurations this is worth a look; for the rest of triangle geometry it is a modest but clean addition. I would bring it to a reading group if the group has anyone interested in named circles, and I would send it to peer review because the result is new, the separation from Tucker is stated cleanly, and the converse adds substance. A referee can check the cocyclicity details in one pass.

Referee Report

0 major / 1 minor

Summary. The paper introduces a new Lemoine-type circle defined via a six-point configuration in a triangle that satisfies a stated cocyclicity criterion. It claims to prove that the points are concyclic, establishes a converse theorem, relates the circle to Bui's Lemoine circle, and proves that the new circle lies outside the Tucker family.

Significance. If the central cocyclicity proof and separation from the Tucker family hold, the work adds a concrete new configuration to the literature on Lemoine circles, with the converse providing a useful characterization. Explicit comparison to Bui's circle and the Tucker family helps delineate the landscape of such circles in triangle geometry.

minor comments (1)

[Abstract] The abstract states that a cocyclicity criterion is satisfied and proved, but does not name the criterion or the six points explicitly; adding one sentence identifying the points (e.g., by their standard triangle-geometry labels) would improve readability without altering the technical content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the paper's main results: the introduction and proof of a new Lemoine-type circle via a six-point cocyclic configuration, the converse theorem, the explicit relation to Bui's Lemoine circle, and the separation from the Tucker family.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper presents a geometric proof establishing a new Lemoine-type circle from a six-point configuration satisfying a stated cocyclicity criterion, together with a converse theorem and explicit relations to prior circles (including Bui's). No equations, fitted parameters, self-definitional reductions, or load-bearing self-citations appear in the abstract or description. The central claims rest on standard Euclidean geometric relations and direct proofs rather than reducing to their own inputs by construction, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review limits visibility; relies on standard Euclidean plane geometry for cocyclicity and circle properties without introducing new parameters or entities.

axioms (1)

standard math Standard axioms of Euclidean plane geometry including properties of circles and concyclic points
Invoked implicitly for the cocyclicity criterion and proof of existence

pith-pipeline@v0.9.0 · 5340 in / 1169 out tokens · 31403 ms · 2026-05-13T18:33:03.839760+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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