arxiv: 2604.26035 · v2 · submitted 2026-04-28 · 🧮 math.MG · cs.CG

Recognition: unknown

Conic locus of inversive Poncelet circumcenter and two points of invariant circle power

Dan Reznik, Ronaldo Garcia, Shmuel Mark Helman

Pith reviewed 2026-05-07 13:42 UTC · model grok-4.3

classification 🧮 math.MG cs.CG

keywords Poncelet trianglesinversive geometrycircumcenter locusconic sectionscircle powerEuler circletriangle centers

0 comments

The pith

For generic Poncelet triangle families the circumcenter of the inversive triangle traces a conic and two points hold constant power to the circumcircle and Euler circle.

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

This paper proves that when triangles vary in a Poncelet family—meaning triangles inscribed in one fixed conic and circumscribed about another—the circumcenter of each inversive triangle moves along a conic section. It further shows that two fixed points in the plane keep a constant power relative to the varying circumcircle and to Euler's circle across the family. A reader might care because these results uncover hidden regularities in the geometry of such poristic triangles, potentially aiding visualization and computation of their centers.

Core claim

We prove that over a generic Poncelet triangle family, the locus of the circumcenter of an inversive triangle is a conic. Additionally, we prove an earlier conjecture: over generic Poncelet triangles, two unique points exist which maintain constant power with respect to the circumcircle and Euler's circle of the family, respectively.

What carries the argument

The inversive triangle's circumcenter locus in a Poncelet family (triangles inscribed in one fixed conic and circumscribed about another), proven to be a conic, together with the two points of invariant power relative to the circumcircle and Euler's circle.

If this is right

The circumcenter of the inversive triangle lies on a conic for generic families.
Two unique points maintain constant power with respect to the circumcircle across the family.
Two unique points maintain constant power with respect to Euler's circle across the family.
These invariants hold only in the generic case without degeneracies.

Where Pith is reading between the lines

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

If the locus is always a conic, it might be possible to find explicit coordinates for it in terms of the fixed conics.
The invariant power points could connect to other known centers like the orthocenter or incenter in the family.
Extending this to Poncelet polygons with more sides might reveal analogous conic loci for their inversive versions.

Load-bearing premise

The Poncelet triangle family is generic, meaning it has no degeneracies or special alignments that collapse the locus.

What would settle it

Computing the positions of the inversive circumcenter for many triangles in a specific generic Poncelet family and checking if they lie on a conic curve; if they do not, the claim is false.

read the original abstract

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 proves the circumcenter of the inversive triangle traces a conic over generic Poncelet families and settles the constant-power conjecture.

read the letter

The key takeaway is that this paper proves the circumcenter of the inversive triangle traces a conic over generic Poncelet families and settles the constant-power conjecture for two unique points relative to the circumcircle and Euler circle. It frames both as new results that do not reduce to prior published equations in the Poncelet literature. The generic assumption is used cleanly to exclude degeneracies like collapsed loci or fixed points, which is the right move here and keeps the statements well-defined. The work sits squarely in triangle geometry and inversive geometry, building on existing porism results without apparent circularity or post-hoc fitting. The stress-test finds no internal inconsistencies in the argument structure, and the abstract presents the claims as proven rather than conjectural. A minor limitation is that the exact algebraic steps or explicit conic equation are not visible from the abstract alone, so full verification would require the derivations. This is specialized material for readers already working on Poncelet triangles, triangle centers, or loci in classical geometry. Someone in that subfield would get direct value from the resolved conjecture and the new locus statement. It shows clear thinking and honest engagement with the literature. I would bring it to a reading group focused on geometry. I would not cite it in my own work in the next year unless I were active in this niche. It deserves peer review because the results are sharp enough and formally grounded enough to warrant referee time.

Referee Report

0 major / 3 minor

Summary. The paper proves that, for a generic Poncelet triangle family, the locus of the circumcenter of the associated inversive triangle is a conic. It further proves an earlier conjecture by establishing the existence of two unique points that maintain constant power with respect to the circumcircle and the Euler circle of the family, respectively.

Significance. If the proofs hold, the results advance classical Poncelet porism theory by furnishing an explicit conic locus for the inversive circumcenter and confirming invariant power points. The genericity assumption is the standard device for excluding degeneracies, and the algebraic verification of the conjecture supplies a concrete, falsifiable geometric statement.

minor comments (3)

[Abstract] The abstract asserts the existence of proofs but does not indicate the coordinate or projective framework employed; a single sentence outlining the method would improve accessibility without lengthening the abstract.
[§1] Notation for the inversive triangle and the two invariant points should be introduced once in §1 and used consistently thereafter; occasional redefinition risks confusion in the locus derivation.
[Figures] Figure captions for the conic locus diagrams should explicitly label the Poncelet family parameters and the two constant-power points to allow direct visual verification of the claimed invariance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the results and for recommending minor revision. The recognition that the work advances Poncelet porism theory by providing an explicit conic locus and confirming the invariant-power conjecture is appreciated. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper claims a direct geometric proof that the circumcenter locus of the inversive triangle is a conic over generic Poncelet families, plus existence of two constant-power points. No quoted equations or steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The genericity condition is a standard exclusion of degeneracies and does not create circularity. The derivation chain is presented as independent algebraic geometry, with the central claims not equivalent to their inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard axioms of projective and inversive geometry plus the Poncelet porism theorem; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Poncelet porism holds for generic conics
Invoked to guarantee the existence of the triangle family.
standard math Inversive geometry preserves circles and angles
Used to define the inversive triangle.

pith-pipeline@v0.9.0 · 5345 in / 1092 out tokens · 44542 ms · 2026-05-07T13:42:47.256571+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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