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REVIEW 4 major objections 6 minor 40 references

Menelaus' and Ceva's theorems for translation triangles in Thurston geometries

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Pith's one-line read The paper proves that Menelaus' and Ceva's theorems hold verbatim for translation triangles in the non-constant-curvature Thurston geometries Nil, Sol, and the universal cover of SL(2,R), using a projection-invariant definition of simple…

desk verdict The Sol simple-ratio definition has a sign error that invalidates the Menelaus theorem; the ~SL2R extension is the sound new part. read the letter →

arxiv 2506.01354 v1 pith:6DKKRYDS submitted 2025-06-02 math.GT math.MG

Jenő Szirmai This is my paper · ORCID
classification math.GTmath.MG MSC 53A2053A3552C3553B20
keywords ThurstongeometriestranslationtrianglesMenelaustheoremCevaNilgeometrySolSL2Rsimpleratio
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

Translation triangles in Nil, Sol, and $\widetilde{\mathrm{SL}_2\mathbb{R}}$ have sides made of translation curves rather than geodesics, and their triangular surfaces were defined in a companion paper. This paper gives each such triangle a signed simple ratio for three points on a translation curve (Definitions 3.13, 3.22, and 3.28) and proves that the classical Menelaus product equals $-1$ and the Ceva product equals $+1$ for these triangles. The interest is that these are classical triangle theorems stated verbatim in geometries of non-constant curvature, going beyond the earlier constant-curvature cases and the $S^2\times\mathbb{R}$/$H^2\times\mathbb{R}$ results. If correct, the same proof scheme should transfer other Euclidean incidence theorems to translation-triangle configurations.

What carries the argument

The carrying object is the signed simple ratio defined on translation curves: in Nil it is the ratio of translation distances $d_t(A,P)/d_t(P,B)$; in Sol it is a formula built from $e^{-d_t\sin\theta}$; in $\widetilde{\mathrm{SL}_2\mathbb{R}}$ it is a ratio of $\tanh$ or $\tan$ of scaled translation distances, depending on the direction parameter $\alpha$. The workhorse is the projection lemmas: each lemma sends the three points on a translation curve to collinear points in a Euclidean model (the $[x,y]$ plane for Nil, the mapped $[x,z]$ plane for Sol, the $x$-axis for $\widetilde{\mathrm{SL}_2\mathbb{R}}$) and shows the non-Euclidean simple ratio equals the Euclidean one. This equality is what converts the curved triangle statement into the classical Menelaus or Ceva theorem; the companion-paper definition of the translation-triangle surface supplies the domain in which the side-curves and cevians live.

What would settle it

Take a concrete Sol translation triangle and a line meeting its three side-curves with one intersection point lying outside its segment; compute both sides of Theorem 3.25 using Definition 3.22 and compare with the Euclidean Menelaus product of the projected points. A mismatch in the exterior-point case would show the 'otherwise' sign branch is wrong; matching in all tested cases would support the reduction.

Watch

Extended reading notes

Core claim

The central claim is that Menelaus' and Ceva's theorems survive unchanged in the curved translation geometry of Nil, Sol, and $\widetilde{\mathrm{SL}_2\mathbb{R}}$, provided the simple ratio of three points on a translation curve is measured with the geometry-specific signed functions introduced in Definitions 3.13, 3.22, and 3.28. For a translation triangle $A_0A_1A_2$, the paper proves $s(A_0,P,A_1)s(A_1,Q,A_2)s(A_2,R,A_0)=-1$ when a line $l$ in the triangle's translation surface meets the three side-curves in $P,Q,R$ (Theorems 3.17, 3.25, and 3.31), and $+1$ when three surface curves through a point $T$ meet the opposite sides in $P,Q,R$ (Theorems 3.16, 3.24, and 3.30). The proofs are reductions: in Nil, fiber projection to the $[x,y]$-plane; in Sol, projection to the $[x,z]$-plane followed by the mapping $(x,z)\mapsto(x,e^{-z})$; in $\widetilde{\mathrm{SL}_2\mathbb{R}}$, projection to the $x$-axis, where translation curves are Euclidean straight lines. In each case the projection lemma (3.14, 3.23, or 3.29) says the simple ratio is unchanged, so the Euclidean theorem applies directly.

Load-bearing premise

The proof stands on the assumption that the signed ratio defined for points outside the segment, especially the Sol formula with its 'otherwise' case, really equals the Euclidean ratio after projection; the Sol argument also assumes the curved surface of the translation triangle, taken from the companion paper, is well enough defined to talk about a line crossing it.

Editorial extensions

If this is right

  • In Nil, Sol, and $\widetilde{\mathrm{SL}_2\mathbb{R}}$, the classical Menelaus and Ceva statements hold verbatim for translation triangles with the simple ratios of Definitions 3.13, 3.22, and 3.28.
  • The simple ratio gives a metric tool for locating where a cevian or transversal cuts a translation side, since it compares translation distances along that side.
  • For $\widetilde{\mathrm{SL}_2\mathbb{R}}$, the proof shows the incidence structure is Euclidean and the geometry enters only through a direction-dependent rescaling of the ratio.
  • Because the method is a reduction to the Euclidean plane, other Euclidean incidence theorems should transfer to translation-triangle configurations by the same projections.
  • The results extend the earlier geodesic-triangle theorems in Nil and the $S^2\times\mathbb{R}$ and $H^2\times\mathbb{R}$ cases to the translation-triangle setting in all three non-constant-curvature Thurston geometries.

Reading between the lines

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

  • A natural stress test is to verify that the Sol simple ratio is independent of the choice of the projected coordinate plane; the paper's proof implicitly assumes invariance of the construction under admissible projections.
  • The ratio-preserving projection idea suggests a uniform framework: define a translation simple ratio in any homogeneous space with a fibred translation structure, then prove Menelaus and Ceva by finding a projection that linearizes the ratio.
  • If the sign convention in Sol's exterior-point case turns out to need adjustment, the fix would likely be a redefinition of the ratio rather than a change in the theorem's statement, since the Euclidean limit is already built into the projection lemma.
  • The method opens a route to define signed lengths and cross-ratios on translation curves, which could connect translation-triangle geometry to the projective geometry of the base plane.
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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

4 major / 6 minor

Summary. The paper proposes generalizations of Menelaus' and Ceva's theorems to translation triangles in the three non-constant curvature Thurston geometries Nil, Sol, and ~SL2R. It defines 'simple ratios' for points on translation curves in each geometry (Definitions 3.13, 3.22, 3.28) and states that the classical product identities hold: product of ratios = 1 for Ceva and = -1 for Menelaus (Theorems 3.16-3.17, 3.24-3.25, 3.30-3.31). The proofs proceed by projecting the configuration onto a coordinate plane and applying the Euclidean theorems, using a map m in the Sol case that converts the projected translation curves into Euclidean segments. The paper relies on the author's companion work [7] for the definition of the translation triangle surface.

Significance. The idea of transplanting Euclidean Menelaus/Ceva via projections and distance-ratio preserving maps is attractive and, for the Nil case, is carried out correctly: projection to the [x,y] plane preserves the translation-distance ratios. If the Sol and ~SL2R parts could be corrected, the paper would offer a uniform method for transferring classical theorems to these geometries. However, as written, the Sol simple-ratio definition contains a sign error that contradicts its main lemma, and the ~SL2R lemma equating the defined ratio with a Euclidean projected ratio is false (see major comments). These are not cosmetic issues; they invalidate Theorems 3.25, 3.30, and 3.31 as stated.

major comments (4)
  1. [§3.4, Definition 3.22 and Lemma 3.23] The 'otherwise' branch in Definition 3.22(1) is incompatible with Lemma 3.23(1). The formula F(A,P,B) = (1 - e^{-d(A,P) sin θ})/(e^{-d(A,P) sin θ} - e^{-d(A,B) sin θ}) is already signed: for P between A and B it is positive, while for P on either exterior ray it is negative. Multiplying by -1 for all exterior points therefore makes every exterior ratio positive. Lemma 3.23(1) asserts s^Sol(A,P,B) = s^E3(A_m,P_m,B_m), which is negative for exterior points under the standard Euclidean sign convention. Menelaus configurations always have an odd number of exterior side intersections, so with the erroneous sign convention Theorem 3.25 would produce product +1 instead of -1. The proof of Theorem 3.25 is only 'similarly to the above proof' and cannot repair this direct contradiction between the definition and the lemma.
  2. [§3.5, Lemma 3.29 and Definition 3.28] Lemma 3.29(1) is false for the simple ratio defined in Definition 3.28. Using the parametrization in Table 1, take α = π/6, λ = 0, A at s = 0, P at s = 1, B at s = 2. Then the x-coordinate is x(s) = tanh(s√cos 2α) sin α / √cos 2α, giving x_A = 0, x_P ≈ 0.4305, x_B ≈ 0.6280. The Euclidean simple ratio for the x-axis projections is (x_P - x_A)/(x_B - x_P) ≈ 2.18, whereas Definition 3.28(1) gives tanh(0.7071)/tanh(0.7071) = 1. Thus the claimed equality with s^E3(A_p,P_p,B_p) does not hold. Since Theorems 3.30 and 3.31 are stated without proof and rely directly on this lemma, they are not established.
  3. [§3.4, Theorems 3.24 and 3.25; also Theorem 3.17] The term 'line' is not defined for the Menelaus statements. In Theorem 3.25, l is assumed to be a line lying in the curved surface S^Sol, and the proof projects it to the [x,z] plane and applies m, but no definition explains why such a line has a projection that becomes a Euclidean straight line after m. Without specifying that l is a connecting curve in the sense of Definition 3.21 (or another precise class of curves), the Menelaus statement is not a well-defined mathematical proposition. The same issue affects Theorem 3.17 in Nil, where 'line' is also used without a definition.
  4. [§3.4, proof of Theorem 3.24] The Ceva proof in the general Sol case is not a proof as written. It asserts that after applying m^{-1} and constructing the curves A_0Q, A_1R, A_2P according to Definition 3.21, 'we get that these curves on S also pass through the point T.' This is exactly the assertion that needs to be justified: for a given T, the cevians A_iT are assumed only to be curves contained in the surface, and the proof does not show that they coincide with the particular connecting curves constructed from the projected Euclidean configuration. Without this identification, the argument is circular. The theorem statement should explicitly require that each cevian is a connecting curve as defined in Definition 3.21, and the proof should verify that the original cevians have that property.
minor comments (6)
  1. [§3.3, Theorem 3.17] The statement contains typos: 'gX_{A1A2}' should be 'g^{Nil}_{A1A2}', and the product includes 'A22' instead of 'A2'.
  2. [§3.4, Theorem 3.25] The statement says 'geodesic triangle' where 'translation triangle' is intended, and 'gS_{A0A2OL}' appears to be a typo for 'g^{Sol}_{A0A2}'.
  3. [§3.4, proof of Lemma 3.23] The notation in the proof is confused: the equality 's^E3_g(A_m, P_m, B_m) = s^E3_g(A_{p,x}, B_{p,x}, P_{p,x})' has the last three points in an order inconsistent with the left-hand side, making the argument difficult to follow.
  4. [§3.5, Theorems 3.30 and 3.31] These theorems have no proof text, only an empty square. Given that the derivations are not immediate from Lemma 3.29, a detailed proof or at least a clear derivation is required.
  5. [§2.2, formula (2.17)] In the case w = 0, the formula gives z(t) = z(0) = 0; this should be stated as z(t) = 0 for all t, since the formula as written could be misread as a constant determined by z(0).
  6. [§3.1] The paper relies on [7] for several properties of the translation triangle surface, but [7] is a submitted manuscript. The author should either include the needed proofs or explicitly list which results from [7] are assumed and how they are used in each proof.

Circularity Check

2 steps flagged · score 6.0 of 10

Sol and Nil theorems reduce to Euclidean results by construction: simple ratios and connecting curves are defined so that the Euclidean theorems transfer verbatim.

  1. self definitional [Section 3.4, Definition 3.21(2), Definition 3.22(1), Lemma 3.23, proof of Theorem 3.24]
    "The Euclidean Ceva theorem is true for this configuration exactly if s^{E3}_g(A^m_0, P^m, A^m_1) s^{E3}_g(A^m_1, Q^m, A^m_2) s^{E3}_g(A^m_2, R^m, A^m_0) = 1. After this we applied to this arrangement, then the m−1 transformation and we construct the A0Q, A1R, A2P ⊂ S^{Sol,t}_{A0A1A2} surface curves as described in Definition 3.21, then we get that these curves on S^{Sol,t}_{A0A1A2} also pass through the point T. This, due to the Lemma 3.23, occurs exactly when the definition of the simple ratio is based on Definition 3.22."

    Definition 3.21 defines the connecting curves as m^{-1}(Euclidean segment) lifted by a cylinder parallel to the y-axis to the triangle surface; Definition 3.22 defines the simple ratio so that Lemma 3.23 equates it with the Euclidean simple ratio of the m-images. The proof first invokes Euclidean Ceva for the m-image and then uses the definition of the connecting curves to assert that the curves pass through T. Therefore the Sol Ceva/Menelaus product is the Euclidean product by construction: the theorem statement is an exact translation of the Euclidean theorem, not an independent Sol-geometric assertion.

  2. renaming known result [Section 3.3, Definition 3.15(2), Lemma 3.14(1), proof of Theorem 3.16]
    "In other cases the connecting curve P1P2 is the image of the translation curve g^{Nil}_{P1P2} into the surface S^{Nil,t}_{A0A1A2} by fibrum projection. ... Moreover, from the Lemma 3.14 follows, that this projection does not change the simple ratio. Thus, the theorem follows directly from the Euclidean Ceva theorem."

    Under fibrum projection (Lemma 3.11) every such connecting curve has a straight Euclidean segment as its image in the [x,y]-plane, and Lemma 3.14 proves s^{Nil}_g(A,P,B) = s^{E3}_g(A_p,P_p,B_p). Hence the Nil Menelaus/Ceva statement for these connecting curves is exactly the Euclidean theorem applied to the projected triangle. The choices of 'connecting curve' and 'simple ratio' are made so that the Euclidean theorem transfers, so the claimed generalization is a relabeling of the Euclidean result rather than a derivation from an independently given Nil notion of line or ratio.

full rationale

The paper's method is an explicit transfer: define 'connecting curves' inside the translation-triangle surface and 'simple ratios' so that a projection or mapping sends the configuration to an Euclidean one, then apply Euclidean Ceva/Menelaus. This is transparent, but it means the central Sol and Nil statements are consequences of the definitions rather than discoveries about pre-existing geometric notions. In Sol (Definitions 3.21 and 3.22, Lemma 3.23) the ratio is defined to equal the Euclidean ratio of m-images and the curves are defined as m-preimages of Euclidean lines, so Theorems 3.24 and 3.25 are Euclidean theorems by construction. In Nil (Definition 3.15, Lemma 3.14) the same holds after fibrum projection. The ^SL2R case is less problematic because translation curves are Euclidean straight lines there. The dependence on the same-authors' submitted manuscript [7] for the translation-triangle surface is load-bearing, but the definition is restated in Section 3.1, so it is not a hidden circularity. Separately, Definition 3.22's 'otherwise' branch flips the sign of an already signed expression, contradicting Lemma 3.23 for exterior points and invalidating Theorem 3.25 as stated; that is a correctness defect, not a circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The central claim rests on several choices that are not independently justified: the surface definition from the submitted paper [7], the tailored simple ratio and connecting curve definitions in Sol, and the implicit convention for signed ratios at exterior points. These choices, rather than an external benchmark, are what make the Sol reduction to Euclidean geometry work.

assumptions (5)
  • standard math Projective models of Nil, Sol, and ~SL2R as described in [10]
    Background models used throughout; taken from Molnar's standard framework.
  • standard math Translation curve parametrizations (2.8), (2.20), Table 1
    Derived from the translation curve differential equations; accepted as definitions in this paper.
  • domain assumption Surface of a translation triangle as defined in [7]
    The paper uses the surface definition from the submitted manuscript [7] without proof or reproduction; the Sol connecting curves depend on this surface.
  • ad hoc to paper Simple ratio and connecting curve definitions in Sol (Definitions 3.21 and 3.22)
    These definitions are chosen so that the projection and mapping m reduce the configuration to a Euclidean one; the sign flip for exterior points is inconsistent with Lemma 3.23.
  • domain assumption Existence of a line l lying in the translation triangle surface for Menelaus in Sol
    Theorem 3.25 assumes such a line without defining it; the surface is curved, so 'line' is ambiguous.
invented entities (3)
  • Translation-like triangular surface S^{X,t}
    purpose: Defines the surface of a translation triangle in Nil, Sol, and ~SL2R
    Introduced in the companion paper [7]; no independent verification is provided in this paper.
  • Connecting curves on the Sol triangular surface (Definition 3.21)
    purpose: Define cevians and transversals within the surface so that the Euclidean reduction works
    Constructed via projection, mapping m, and cylinder intersection; chosen to force the theorems.
  • Fibre type and general type translation triangles
    purpose: Classify triangles for the proofs
    A classification convention, not an independently evidenced entity.

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Cite this review

Pith. "Pith review of Menelaus' and Ceva's theorems for translation triangles in Thurston geometries." pith.science (2026). https://pith.science/paper/6DKKRYDS

@misc{pith2026250601354,
  author       = {Pith},
  title        = {Pith review of: Menelaus' and Ceva's theorems for translation triangles in Thurston geometries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6DKKRYDS}},
  note         = {Machine review of arXiv:2506.01354}
}
read the original abstract

After having investigated and defined the ``surface of a translation-like triangle" in each non-constant curvature Thurston geometry \cite{Cs-Sz25}, we generalize the famous Menelaus' and Ceva's theorems for translation triangles in the mentioned spaces. The described method makes it possible to transfer further classical Euclidean theorems and notions to Thurston geometries with non-constant curvature. In our work we will use the projective models of Thurston geometries described by E. Moln\'ar in \cite{M97}.

Figures

Figures reproduced from arXiv: 2506.01354 by the authors.

Figure 1
Figure 1. Translation triangle with vertices A0 = (1, 0, 0, 0), A1 = (1, −1, 1, 1), A2 = (1, 1/2, 1, 1/2) and its projected image by fibers into [x, y] coordinate plane. Definition 3.13 If A, B and P are distinct points on a translation curve in the Nil space, then their simple ratio is s Nil g (A, P, B) = d Nil,t(A, P)/dNil,t(P, B), if P is between A and B, and s Nil g (A, P, B) = −d Nil,t(A, P)/dNil,t(P, B), (see [PITH_FUL… view at source ↗
Figure 2
Figure 2. Translation triangle with vertices A0 = (1, 0, 0, 0), A1 = (1, 2, 0, 3), A2 = (1, −1, 0, 2) lying in the [x, z] coordinate plane. Proof: 1. From the Lemma (3.11) it is seen that the points Ap , Bp and P p are lo￾cated on a straight line in the [x, y] plane. From the formulas (2.8), (2.9) it is directly seen that the translation distances on straight line ApBp are equal to the Euclidean distances and that the ratio o… view at source ↗
Figure 3
Figure 3. Nil translation curve segment g Nil A,B with point P ∈ g Nil A,B and its pro￾jected image by fibers into [x, y] coordinate plane. Theorem 3.16 (Ceva’s theorem for translation triangles in Nil space) If T is a point not on any side of a translation triangle A0A1A2 in Nil such that the curves A0T and g Nil A1,A2 meet in Q, A1T and g Nil A0A2 in R, and A2T and g Nil A0A1 in P, (A0T, A1T, A2T ⊂ SNil A0A1A2 ) then s Nil … view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Translation triangle with vertices A0 = (1, 0, 0, 0), A1 = (1, 5/4, 1/2, 1), A2 = (1, 1/5, 1, 1/2) and its image by orthogonal projection into [x, z] coordinate plane [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: The projected translation triangle with vertices [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Translation triangle with vertices A0 = (1, 0, 0, 0), A1 = (1, 1/2, 3/4, 0), A2 = (1, 2/3, 1/4, −1/3) (left) The translation segment g SL^2R.t A,P,B and its image by orthogonal projection into x axis. The following lemma is a direct consequence of Definition 3.28. and …

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.