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REVIEW 3 major objections 5 minor 13 references

More Relationships between a Central Quadrilateral and its Reference Quadrilateral

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper catalogs, for 28 quadrilateral shapes and 1000 triangle centers, exactly how the quadrilateral formed from the four half-triangle centers relates to the original—congruence, similarity, area, circles, and more.

desk verdict A useful, honestly-reported extension of a computer-discovered geometry catalog, but the proof-status convention breaks down in §32 and needs fixing before it can be used as a reliable reference. read the letter →

arxiv 2506.17240 v1 pith:T662IBN7 submitted 2025-06-05 math.GM

Stanley Rabinowitz , Ercole Suppa This is my paper
classification math.GM MSC 51M0451-08
keywords centralquadrilateralhalftrianglestrianglecentersgeometrycomputer-discoveredmathematicsbarycentriccoordinatescyclicquadrilateralshomothety
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to map, systematically, how a quadrilateral relates to the quadrilateral formed when the same triangle center is placed in each of its four half-triangles, the triangles cut off by the diagonals. For 28 quadrilateral shapes and the first 1000 indexed triangle centers, it checks whether the two quadrilaterals are congruent, similar, homothetic, perspective, area- or perimeter-related, or share notable circles, conics, and centers. It reports dozens of exact relationships, including that centroid centers always give a homothetic copy at scale $1/3$, orthocenter centers always preserve area, and for cyclic quadrilaterals the $X_{399}$ centers give concentric circumcircles with radius ratio $1:2$. Many entries are proved by exact barycentric-coordinate computation; those verified only numerically are marked. The value of the enterprise is a structured catalogue of center-by-shape behavior that states new geometry facts and points to families of centers yet to be explained.

What carries the argument

The construction is the half-triangle device: the diagonals of a quadrilateral $ABCD$ divide it into four triangles $\triangle BCD$, $\triangle ACD$, $triangle ABD$, and $\triangle ABC$; placing the same indexed center $X_n$ in each gives points $E,F,G,H$, and the quadrilateral $EFGH$ is the central quadrilateral. The proof machinery is barycentric coordinates with $\triangle ABC$ as reference and $D=(p:q:r)$; each quadrilateral shape becomes an algebraic condition on $a,b,c,p,q,r$ (cyclicity, for example, is $a^2qr+b^2pr+c^2pq=0$), and each center's coordinates come from its center function. Claims are checked by exact symbolic simplification under these constraints, with numerical computer search as the discovery engine and, when no exact proof was found, as the only evidence. The named identities that carry the arguments include the Euler-Poncelet point, the common point of the nine-point circles of the half-triangles, and the Gergonne-Steiner point.

What would settle it

Pick any entry colored red in the tables, write out the two quadrilaterals' coordinates in the barycentric system of Section 5, and compute the claimed equality as a rational function of the side lengths and shape parameters; if that function is not identically zero under the shape condition, the entry is false.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that for each of the 28 listed quadrilateral classes and each of the first 1000 indexed triangle centers, the relationship between the reference quadrilateral $ABCD$ and the central quadrilateral $EFGH$ is a definite geometric fact, and the paper's tables state which of the tested relationships hold for which centers. The flagship proved cases are: $X_2$ (centroid) yields similarity with ratio $3$, homothety at the centroid, and area ratio $9:1$ for any quadrilateral; $X_4$ (orthocenter) preserves area and gives a common rectangular-hyperbola circumconic with center at the Euler-Poncelet point; $X_{399}$ on a cyclic quadrilateral gives concentric circumcircles with radii in ratio $1:2$; and $X_1$ on a tangential quadrilateral makes $ABCD$ and $GHEF$ perspective at the incenter. For cyclic quadrilaterals, centers lying on the reference triangle's circumcircle automatically give the same circumcircle. Where the authors could not provide a synthetic or symbolic proof, they mark the center red and rely on 15-digit numerical checks; the paper is explicit that these are discoveries, not proofs.

Load-bearing premise

The unproved table entries depend on the assumption that numerical checks to 15 digits, plus visual variation of a diagram as points move, are reliable signs that an exact relationship holds for every quadrilateral of that type.

Editorial extensions

If this is right

  • For any quadrilateral, the centroid-centered central quadrilateral is a $1/3$-scale homothetic copy centered at the original quadrilateral's centroid, so the area is divided by $9$.
  • For any quadrilateral, the orthocenter-centered central quadrilateral has the same area as the original, even though its shape generally differs.
  • For a cyclic quadrilateral, any triangle center that lies on the circumcircle of the reference triangle produces the same circumcircle for the central quadrilateral, which transfers a whole list of centers into quadrilateral statements.
  • For a tangential quadrilateral, the four lines from the vertices to the incenters of the opposite half-triangles concur at the incenter, giving a new perspectivity.
  • The red, numerically verified table entries are asserted as conjectured geometric facts; if they hold exactly, they extend the catalogue of center-to-quadrilateral relations well beyond the proved cases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step the paper does not take is to treat the parameter $k$ in families like $\cos B\cos C+k\cos A$ as a continuous variable and compute the central quadrilateral's invariants as rational functions of $k$, turning isolated center entries into full continuous families.
  • Because the search only tested rational area ratios with denominators below 10, relationships with ratios such as $7/13$ or with expressions involving square roots would have been missed; a symbolic-area-search extension would likely reveal further table entries.
  • The same half-triangle construction could be applied to triangle points that are not triangle centers, such as the Brocard points the paper tests for squares, suggesting a broader theory of 'central figures' beyond the first 1000 indexed centers.
  • The red entries form a concrete testbed for automated theorem proving: each is a single exact algebraic identity waiting to be confirmed or refuted by symbolic computation, independent of the numerical evidence that found it.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies central quadrilaterals: for a reference quadrilateral ABCD, the four triangles formed by its sides and diagonals (half-triangles) host the same triangle center X_n, and the four resulting points E,F,G,H form the central quadrilateral EFGH. For 28 quadrilateral shapes and the first 1000 triangle centers, the authors use the GeometricExplorer program to detect relationships such as congruence, similarity, equal area, equal perimeter, common circumconics, perspectivity, and homothety between ABCD and EFGH. Some results are proved exactly via barycentric coordinates (e.g., Theorems 5.1, 6.1, 7.1, 14.4, 14.5), while many others are supported only by 15-digit numerical computation. The paper's Section 5 states that numerical evidence is not proof and that unproved results are colored red in tables, but this convention is not applied consistently, particularly in Section 32 where several statements labeled 'Theorem' are only computer discoveries and are followed by open questions asking for proofs. The paper is an extension of the authors' earlier work [4,5] and includes many tables of newly found relationships.

Significance. If the catalog of relationships is reliable, this is a useful contribution to computer-discovered geometry: it provides exact symbolic proofs for several theorems (e.g., Theorem 5.1, 7.1, 14.4, 14.5), a reproducible computational search methodology, and a systematic map of how central quadrilaterals behave across a wide range of triangle centers and quadrilateral shapes. The paper is also commendably explicit about the difference between numerical evidence and proof in its methodology section. However, the central claim that the listed relationships are genuine geometric facts is undermined by the inconsistent application of that very distinction: several 'Theorem' statements in Section 32 rest solely on computer search, and the tables in Sections 6-31 do not visibly carry the red/proved status markers described in Section 5. The paper also leaves undefined the notions of area and perimeter for the non-convex or self-intersecting central quadrilaterals that it explicitly allows. These issues are load-bearing for the catalog's usability and must be repaired before the results can be reliably used as a reference.

major comments (3)
  1. [§5 vs §32] The proof-status convention stated in Section 5 (numerical evidence is not proof; unproved results are colored red) is not applied in Section 32. Theorems 32.2, 32.3, 32.4, 32.6, 32.7, 32.8, and 32.9 are all presented as 'Theorem' but are introduced as 'found by computer' with no proof, no red marking, and no pointer to a symbolic notebook. Several of these are immediately followed by Open Questions 9, 11, and 13 asking for 'purely geometrical' proofs, confirming that no analytic proof is supplied. Because the reader cannot distinguish established theorems from numerical conjectures within the same document, the central claim that the listed relationships are genuine geometric facts is not reliably supported. The authors should either provide exact symbolic proofs or explicit notebook references for every statement labeled Theorem, or re-label these results as conjectures/computer discoveries and apply the color convention consistently.
  2. [§1, §4, §6] The paper does not define 'area' and 'perimeter' for the central quadrilateral EFGH, although it explicitly states in Section 1 that EFGH need not be convex. The relationships table in Section 4 includes entries such as '[ABCD] = k[EFGH]' and '∂ABCD = ∂EFGH', and theorems such as Theorem 6.2 ([ABCD] = 9[EFGH]) and Theorem 6.6 ([ABCD] = [EFGH]) assert exact area equalities. If EFGH is concave or self-intersecting, the shoelace formula yields a signed area while the usual polygon area is the absolute value, and these give different values for self-intersecting cases. The perimeter of a self-intersecting quadrilateral also needs a convention (e.g., sum of side lengths of the polygonal path, or perimeter of the convex hull). The truth of the area and perimeter claims can depend on these choices, so the paper must specify precisely how these quantities are computed.
  3. [§6–§31 tables] The red/proved status marker described in Section 5 is not visible in the tables as presented in the text. For example, the Central Quadrilaterals of Cyclic Quadrilaterals tables in Section 14 list centers such as 5, 550, 376, 140, etc. for area ratios and other relationships, but there is no per-entry indication of whether the entry is a proved theorem, an analytic proof from a notebook, or a purely numerical discovery. The legend in Section 5 says numerically verified centers are colored red and proved centers are not, but no colors are reproduced in the manuscript text. This is not merely a visual issue: it is the mechanism by which the paper itself says a reader should distinguish reliable from conjectural results. The authors should add an explicit per-entry marker (e.g., bold for proved, red for numerical, or a footnote referencing the specific notebook) so that each table entry carries its proof status.
minor comments (5)
  1. [§4] The footnote that only rational area ratios k with denominators less than 10 were checked is a significant limitation and should be stated in the Introduction or Abstract, since later sections report ratios such as 144/25 and 100/9 (Section 14) and 25/4 (Section 30).
  2. [§32.6] In the list of n for which a cyclic quadrilateral has a common non-circular circumconic, the number '11738' appears twice; one occurrence is likely a typo for another ETC index such as 1173, and the list should be corrected.
  3. [§5.1] For reproducibility, the example proof should state the exact version of the baricentricas.m package used and include the precise Mathematica commands that generate the final simplification result, since the displayed code shows the definitions of the routines but not the full session output.
  4. [§32.2] The term 'orthoptic quadrilateral' is defined only in the future-work section (32.2); if it is a new class of quadrilaterals, it should be defined and listed in the table of quadrilateral types in Section 2 rather than introduced informally in a later section.
  5. [References] Some references are informal, such as [6] ('personal correspondence') and [13] ('submitted'); for any theorem attributed to such a source, a precise statement or proof should be included in the paper to keep the catalog self-contained.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the catalog is produced by independent computer search and symbolic barycentric verification, with only minor reliance on the authors' prior proved results.

full rationale

The derivation chain is not circular. The central quadrilateral is a geometric construction defined independently of the claims, and the claimed relationships (similarity, area ratios, homothety, perspectivity, common conics) are nontrivial outputs of a search over the first 1000 triangle centers and 28 quadrilateral shapes, not parameters fitted to force the conclusions. Section 5 explicitly states that 15-digit numerical checks are not proofs and that unproved entries are colored red; symbolic barycentric proofs are supplied for many statements (e.g., Theorem 5.1 and the analytic proof of Theorem 14.4 in the supplementary material). The paper does cite the authors' own prior papers [4] and [5] for several base theorems (Theorem 6.1, Theorem 6.6, Theorem 30.1, and part of Theorem 23.1), but those are independently published with proofs and are not used to fit the new catalog entries, so this is minor self-citation rather than a circular reduction. The main caveat is epistemic, not circular: §5's red-coloring rule is not consistently applied in §32, where Theorems 32.2, 32.3, 32.4, 32.6–32.8, and 32.9 are labeled 'Theorem' but are described only as 'found by computer' with no red marking or supplied analytic proof, and Open Questions 9, 11, and 13 ask for 'purely geometrical' proofs. That is an inconsistency in proof-status labeling and a correctness risk, but it does not make any claimed result equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central claims rely on standard Euclidean geometry and barycentric coordinates, the correctness of ETC center definitions and coordinates, and the reliability of the computer algebra and numerical search tools. No new free parameters or invented entities are introduced.

assumptions (5)
  • standard math Standard axioms of Euclidean geometry and barycentric coordinates.
    The paper works entirely in the Euclidean plane and uses barycentric coordinates with respect to triangle ABC.
  • domain assumption The coordinates and definitions of the 1000 triangle centers X_n are taken as correct from the Encyclopedia of Triangle Centers [2].
    Section 3 defines triangle centers via Kimberling's center functions, and the paper relies on ETC coordinates for all X_n used in the computer search and proofs.
  • domain assumption The baricentricas.m Mathematica package performs exact symbolic barycentric computations correctly.
    Section 5 states that analytic proofs are performed with Mathematica and the baricentricas.m package; any bug there would invalidate the proven theorems.
  • domain assumption The 15-digit numerical results from GeometricExplorer reliably indicate exact geometric relationships.
    Section 5 acknowledges numerical computation is not proof, but the paper still reports many unproven relationships as results, implicitly relying on this assumption.
  • domain assumption The quadrilateral shape conditions (e.g., orthodiagonal, cyclic, tangential) are correctly translated into algebraic constraints on the barycentric coordinates p, q, r.
    Section 5 provides a table of analytic conditions; an error in these translations would make the shape-specific results invalid.

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Pith. "Pith review of More Relationships between a Central Quadrilateral and its Reference Quadrilateral." pith.science (2026). https://pith.science/paper/T662IBN7

@misc{pith2026250617240,
  author       = {Pith},
  title        = {Pith review of: More Relationships between a Central Quadrilateral and its Reference Quadrilateral},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T662IBN7}},
  note         = {Machine review of arXiv:2506.17240}
}
read the original abstract

The diagonals of a quadrilateral form four associated triangles, called half triangles. Each half triangle is bounded by two sides of the quadrilateral and one diagonal. If we locate a triangle center (such as the incenter, centroid, orthocenter, etc.) in each of these triangles, the four triangle centers form another quadrilateral called a central quadrilateral. For each of various shaped quadrilaterals, and each of 1000 different triangle centers, we compare the reference quadrilateral to the central quadrilateral. Using a computer, we determine how the two quadrilaterals are related. For example, we test to see if the two quadrilaterals are congruent, similar, have the same area, or have the same perimeter.

Figures

Figures reproduced from arXiv: 2506.17240 by the authors.

Figure 1
Figure 1. Half Triangles The triangles have been numbered so that triangle 1 is opposite vertex A, triangle 2 is opposite vertex B, etc. The four triangles are △BCD, △ACD, △ABD, and △ABC. Triangle centers are selected in each triangle (for example, incenters, centroids, or orthocenters). The same type of triangle center is used with each half triangle. In order, the names of these points are E, F, G, and H, as shown in [PITH… view at source ↗
Figure 2
Figure 2. Central Quadrilateral The purpose of this paper is to determine interesting relationships between a ref￾erence quadrilateral and its central quadrilateral. This paper extends our previous results found in [5] [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Quadrilateral Shapes 3. Centers In this study, we will place triangle centers in the four half triangles. We use Clark Kimberling’s definition of a triangle center [1]. A center function is a nonzero function f(a, b, c) homogeneous in a, b, and c and symmetric in b and c. Homogeneous in a, b, and c means that f(ta, tb, tc) = t n f(a, b, c) for some nonnegative integer n, all t > 0, and all positive real numbers (a, … view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: barycentric coordinate system for quadrilateral ABCD The barycentric coordinates for the various triangle centers were found from [2]. To find the coordinates of a center (u : v : w) with respect to a triangle XY Z, we use the function CentroETCTriangulo in the baricen…
Figure 5
Figure 5. Figure 5: orthodiagonal quad with X5-points =⇒ m[ABCD] = dp[EF GH] Note that W, X, Y , and Z are the midpoints of the sides of quadrilateral ABCD, making O the centroid. We need to show that O coincides with the intersection of diagonals EG and F H of quadrilateral EF GH. Proof.…
Figure 6
Figure 6. Figure 6: general quadrilateral with X2-points =⇒ ABCD ∼ EF GH Theorem 6.2. Let ABCD be an arbitrary quadrilateral. Let E, F, G, and H be the X2-points of △BCD, △CDA, △DAB, and △ABC, respectively. Then [ABCD] = 9[EF GH] ( [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: general quadrilateral with X2-points =⇒ homot(ABCD, EF GH) Proof. From [8] we know that the lines from the vertices of a triangle to the centroid of the opposite half triangle meet in a point known as the centroid of the quadrilateral. Thus, the quadrilaterals are pers…
Figure 8
Figure 8. Figure 8: general quadrilateral with X4-points =⇒ [ABCD] = [EF GH] The following result comes from [9]. Theorem 6.7. Let ABCD be an arbitrary quadrilateral. Let E, F, G, and H be the X4-points of △BCD, △CDA, △DAB, and △ABC, respectively. Then quadrilaterals ABCD and EF GH have a…
Figure 9
Figure 9. Figure 9: general quadrilateral with X4-points =⇒ hyperb(ABCD, EF GH) Corollary 6.8. Let ABCD be an arbitrary quadrilateral. Let E, F, G, and H be the X4-points of △BCD, △CDA, △DAB, and △ABC, respectively. Then ponce[ABCD] = ponce[EF GH]. 6.4. Properties involving X5. Theorem 6.…
Figure 10
Figure 10. Figure 10: tangential quadrilateral with X1-points =⇒ persp[ABCD, GHEF] Proof. The point G is the incenter of △ABD, hence G lies on the angle bisector of ∠BAD. Thus, G ∈ AI. Similarly, H ∈ BI, E ∈ CI, and F ∈ DI. Therefore, AG, BH, CE, and DF concur in I. Hence, quadrilaterals A…
Figure 11
Figure 11. Figure 11: orthodiagonal quad with X4-points =⇒ dp(ABCD) = dp(EF GH) Proof. Let P be the diagonal point of quadrilateral ABCD. Since ABCD is or￾thodiagonal, AP is an altitude of △ABD. Since E is the orthocenter of △BCD, E lies on this altitude. Similarly, G also lies on this alt…
Figure 12
Figure 12. Figure 12: cyclic quad with X399-points =⇒ concentric[1 : 2](ABCD, EF GH) An analytic proof of Theorem 14.4 is given in the supplementary material accom￾panying the on-line publication of this paper. Theorem 14.5. Let ABCD be a cyclic quadrilateral. Let X be any triangle center …
Figure 13
Figure 13. Figure 13: kite =⇒ kite Proof. From Theorem 6.27 of [4], the central quadrilateral EF GH is a kite with EF = EH and GF = GH ( [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: shows the case when n = 46. Because of this theorem, to prove that the reference quadrilateral and the central quadrilateral have the same perimeter (∂[ABCD] = ∂[EF GH]), it is only necessary to prove that AB+BC = EF +F G [PITH_FULL_IMAGE:figures/full_fig_p027_14.png]
Figure 15
Figure 15. Figure 15: cyclic quadrilateral with X1173-points =⇒ conic[ABCD, EF GH] [PITH_FULL_IMAGE:figures/full_fig_p028_15.png]
Figure 3
Figure 3. Figure 3: There are many other shapes of quadrilaterals. Study these other [PITH_FULL_IMAGE:figures/full_fig_p029_3.png]
Figure 16
Figure 16. Figure 16: orthoptic quadrilateral with X6-points =⇒ persp[ABCD, EF GH] Open Question 9. Is there a purely geometrical proof of this result? An orthocentric quadrilateral is a quadrilateral in which each vertex is the ortho￾center of the triangle formed by the other three vertic…
Figure 17
Figure 17. Figure 17: orthocentric quad with X11-points =⇒ m[ABCD] = o[EF GH] Open Question 10. Is there a purely geometrical proof of this result? [PITH_FULL_IMAGE:figures/full_fig_p029_17.png]
Figure 18
Figure 18. Figure 18: trapezoid with X3-points =⇒ ortho(ABCD, EF GH) Open Question 11. Is there a center, X, such that quadrilaterals ABCD and EF GH have a common inconic? Open Question 12. Is there a center, X, and a tangential quadrilateral ABCD, such that the central quadrilateral EF GH…
Figure 19
Figure 19. Figure 19: square with Brocard points =⇒ [ABCD] = 5[EF GH] Open Question 13. Is there a purely geometrical proof of this result? 32.9. Place different centers in different half triangles. Would we find any interesting results if we place Xn-points in triangles ABC and ACD, but p…

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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    https://www.jstor.org/stable/2690608

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