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REVIEW 3 major objections 4 minor 6 references

Relationships Between Six Incircles

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read When P is the circumcenter, alternating reciprocal inradii balance exactly.

desk verdict Theorem 5.1 is a clean, fully proved new result, but several other claimed identities rest on omitted CAS details that a referee should require before acceptance. read the letter →

arxiv 1908.02151 v2 pith:SC6DVQ65 submitted 2019-08-04 math.HO math.MG

Stanley Rabinowitz This is my paper · ORCID
classification math.HOmath.MG MSC 51M0451-04
keywords Euclideangeometrytriangleincirclesinradiicevianscircumcentercentroidorthocenter
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

This paper studies the six small triangles created when a point $P$ inside a triangle $ABC$ is joined to the vertices, and asks how the radii of their incircles relate. Its central new result is that when $P$ is the circumcenter the alternating sum of reciprocals vanishes: $1/r_1 + 1/r_3 + 1/r_5 = 1/r_2 + 1/r_4 + 1/r_6$. The paper also records product and reciprocal-sum identities for the orthocenter and centroid, and more complex polynomial identities for special incenter configurations and one-parameter families of points. These identities matter because they show that the six inradii, which look independent, are constrained by the geometry of the point $P$ in a way that depends only on a few local angles.

What carries the argument

The key mechanism is Lemma 5.1, a local cotangent decomposition: if $P$ is the circumcenter and $R$ is the circumradius, then for each small triangle the parent radius is split into $R = r_1(\cot\alpha + \cot(\beta/2))$, so $R/r_1 = \cot\alpha + \cot(\beta/2)$, where $\alpha$ is the base angle at $A$ and $\beta$ is the angle at $B$ along the cevian. Writing the analogous six equations, the alternating sum of the reciprocal inradii becomes a sum of cotangent terms that cancels pairwise. For the centroid case, the machinery is different: the six small triangles have equal area and the semiperimeters satisfy $s_1+s_3+s_5 = s_2+s_4+s_6$, so the identity $r=K/s$ converts that semiperimeter balance into the reciprocal-sum identity.

What would settle it

Compute the six inradii for a triangle with $P$ at the incenter and $\angle B = 120^\circ$, substitute into $r_1r_2r_3+r_3r_4r_5+r_3r_4r_6 = r_1r_3r_4+r_2r_3r_4+r_4r_5r_6$, and check that the difference is zero; a nonzero value disproves Theorem 6.2. Equivalently, evaluate any random scalene triangle with $P$ at the circumcenter and test $1/r_1+1/r_3+1/r_5 = 1/r_2+1/r_4+1/r_6$.

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Extended reading notes

Core claim

The paper's load-bearing new discovery is Theorem 5.1: for any triangle, if $P$ is the circumcenter, then $1/r_1 + 1/r_3 + 1/r_5 = 1/r_2 + 1/r_4 + 1/r_6$. The proof decomposes the circumradius $R$ along a cevian into two pieces involving cotangents of the local angles; the six equations then cancel in alternating pairs. Around this, the paper assembles a family of analogous results: for the orthocenter, $r_1r_3r_5 = r_2r_4r_6$; for the centroid, the same reciprocal-sum identity as the circumcenter, plus $R_1R_3R_5=R_2R_4R_6$; for the incenter under special angle conditions, polynomial identities such as $r_1r_2r_3+r_3r_4r_5+r_3r_4r_6 = r_1r_3r_4+r_2r_3r_4+r_4r_5r_6$; and for arbitrary $P$, the area identities $K_1K_3K_5=K_2K_4K_6$ and $1/K_1+1/K_3+1/K_5=1/K_2+1/K_4+1/K_6$. The special-angle identities are verified by computer algebra, with several proofs omitted.

Load-bearing premise

The load-bearing premise is that the symbolic algebra simplifications omitted from Theorems 6.2, 7.2, and 7.3 are exactly correct; if any of those unshown reductions contains an error, those specific identities fail.

Editorial extensions

If this is right

  • For any triangle, if Theorem 5.1 is correct, the identity $1/r_1 + 1/r_3 + 1/r_5 = 1/r_2 + 1/r_4 + 1/r_6$ holds at the circumcenter and is invariant under scaling and rotation.
  • The classical centers pair up: the orthocenter gives a product identity, while the centroid and circumcenter give the same reciprocal-sum identity, showing that the algebraic form of the relation encodes which center is chosen.
  • For the $30^\circ$--$60^\circ$--$90^\circ$ incenter configuration, the six inradii have explicit radical closed forms, so the five displayed identities among them can be checked exactly, not merely numerically.
  • If the omitted computations in Theorems 7.2 and 7.3 are valid, there are one-parameter families of interior points, not just classical centers, for which alternating reciprocal-sum identities hold.
  • Taken with the area identities $K_1K_3K_5=K_2K_4K_6$ and $1/K_1+1/K_3+1/K_5=1/K_2+1/K_4+1/K_6$, the radius identities suggest that the alternating pattern is a general feature of the cevian configuration, with the semiperimeter deciding when it passes from areas to radii.

Reading between the lines

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

  • Generalizing the cotangent cancellation in Lemma 5.1 suggests that any point $P$ whose six local angles pair to equal alternating sums will also satisfy the reciprocal-sum identity; the circumcenter is one such point. The paper does not state this characterization.
  • The explicit radical values for the $30^\circ$--$60^\circ$--$90^\circ$ incenter case give a ready-made numerical testbed for conjectures about other points $P$; searching for points where the displayed linear or polynomial identities hold could reveal more families, as the paper's open question about straight-line loci hints.
  • The absence of one-parameter families for functions other than $1/r$, which the paper reports, is itself a clue: reciprocal inradii, not radii or squared radii, are the natural coordinates for this configuration, possibly because they linearize the angle relations.
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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 / 4 minor

Summary. The paper studies the six small triangles obtained by drawing the cevians through an interior point P of a triangle ABC, and the inradii r1,...,r6 of their incircles. It proves several identities: for P the orthocenter (r1r3r5 = r2r4r6), for P the centroid (1/r1 + 1/r3 + 1/r5 = 1/r2 + 1/r4 + 1/r6), and a new theorem for P the circumcenter (Theorem 5.1, the same reciprocal-sum identity). It also states special-case identities when P is the incenter and when P is defined by specified angles, and it closes with open questions. The central new result, Theorem 5.1, is proved by a self-contained geometric argument; several of the additional identities are asserted with proofs that say 'the details are omitted' or rely on unshown symbolic algebra simplifications.

Significance. If all claims are correct, the paper makes a pleasing and useful contribution to elementary triangle geometry. The proof of Theorem 5.1 is genuinely elegant and fully self-contained, and it does not depend on the computational steps used elsewhere. The special-angle identities in Sections 6 and 7 are exact, checkable statements that would be of interest to the sangaku/triangle-geometry community. The paper is also honest about its computational discovery procedure and its open questions. However, the load-bearing exact identities beyond Theorem 5.1 are not actually verified in the manuscript: several proofs are explicitly omitted, and no machine-readable check or detailed algebraic certificate is supplied. The central theorem is sound, but the paper as it stands requires the reader to accept a substantial amount of unshown symbolic algebra.

major comments (3)
  1. [Section 6, Theorem 6.2] The proof of Theorem 6.2 consists of the sentence 'This theorem can be proven using the same procedure that was used to prove Theorem 6.1. The details are omitted.' The theorem is a non-obvious polynomial identity in the six inradii for the one-parameter family with angle ABC = 120 degrees, so the missing derivation is load-bearing. The authors should supply the explicit expressions for r1,...,r6 in terms of the remaining angle and show the reduced identity after substitution, or provide an exact symbolic algebra certificate (for example, a notebook or a file with the full simplification).
  2. [Section 7, Theorems 7.2 and 7.3] The text states that the proofs of Theorems 7.2 and 7.3 are 'similar to the proof of Theorem 7.1' and that 'the details are omitted.' Moreover, the discovery procedure described later in Section 7 used FindIntegerNullVector on data sampled at integral-degree angles, which is heuristic and cannot by itself prove an identity for arbitrary parameter t. Since these theorems claim exact one-parameter families, the authors need to provide a complete derivation or at least a verification that works at the symbolic level for arbitrary t, rather than for finitely many sampled angles.
  3. [Section 6, Theorem 6.1] In the proof of Theorem 6.1 all segment lengths are expressed in terms of a, b, c, and the reduction to a function of c alone is described as 'Simplifying this expression (using a symbolic algebra system), we find that the result is 0.' No simplified expression is displayed. This is a local gap rather than a fatal one, but it would substantially improve the paper to show the reduced expression after substituting a = pi/2 - b - c and b = pi/6, or to include a CAS verification file for this step as well.
minor comments (4)
  1. [Section 8] There is a typo in the opening sentence of Section 8: 'cirumcircle' should be 'circumcircle.'
  2. [References] In reference [6], the French title 'Transformation des propiétés métriques des figures' appears to contain a typo; it should be 'propriétés.'
  3. [Section 6, Theorem 6.3] The exact radical values of r1,...,r6 in Theorem 6.3 are stated without derivation, and the verification is described only as substitution with 'computer simplification, as necessary.' Since this is a concrete example rather than a general theorem, this is acceptable, but a short explanation of how the values were obtained would be helpful.
  4. [Section 2, Figure 2] The numbering convention for the six triangles and their incircles is clear in the text but would be easier to follow if Figure 2 explicitly labeled the incenter of one triangle, such as X in the proof of Lemma 5.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the main derivations are self-contained geometric arguments.

full rationale

The paper's central new result, Theorem 5.1, is proved from Lemma 5.1, which expresses R/r_i as a sum of cotangents based on angle bisectors and right-triangle trigonometry; adding the six displayed equations exhibits the desired identity without assuming it. Earlier results for the orthocenter and centroid are likewise proved from elementary similarity, area, and semiperimeter arguments, with external attributions that are not load-bearing. The incenter and special-point theorems are discovered using numerical search (FindIntegerNullVector) and then asserted to be verified by symbolic algebra, with some details omitted; this is an evidentiary gap in the verification of those identities, not circular reasoning, because the radii are computed from independent side-length and area formulas rather than defined in terms of the target equations. No step in the paper reduces a claimed prediction to a fitted parameter, a self-citation chain, or a definitional equivalence involving the radii under study. Accordingly, the paper merits a circularity score of 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest only on standard Euclidean theorems and on the assumed correctness of computer algebra simplifications; no free parameters or invented entities are introduced.

assumptions (4)
  • standard math Standard Euclidean plane geometry axioms, including parallel postulate and area formulas.
    All constructions and proofs are in the Euclidean plane; no alternative geometry is considered.
  • standard math Extended Law of Sines, Law of Cosines, Ceva's theorem, and Menelaus's theorem are valid.
    Invoked in Lemmas 5.1, 6.1, and 7.1; these are classical theorems assumed without proof.
  • domain assumption The point P is strictly inside the triangle, so all six cevians and triangles are nondegenerate.
    The setup and positive inradii require P to be an interior point; the paper states this at the start of Section 2.
  • domain assumption The symbolic algebra system (e.g., Mathematica) performs exact simplification without error.
    Theorems 6.1, 6.2, and 7.1 through 7.3 rely on simplifying trigonometric or polynomial expressions that are not shown in the paper; no system version or output is provided.

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Cite this review

Pith. "Pith review of Relationships Between Six Incircles." pith.science (2026). https://pith.science/paper/SC6DVQ65

@misc{pith2026190802151,
  author       = {Pith},
  title        = {Pith review of: Relationships Between Six Incircles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SC6DVQ65}},
  note         = {Machine review of arXiv:1908.02151}
}
read the original abstract

If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into six smaller triangles. We give theorems about the relationship between the radii of the circles inscribed in these triangles.

Figures

Figures reproduced from arXiv: 1908.02151 by the authors.

Figure 1
Figure 1. 1This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. 51 arXiv:1908.02151v2 [math.HO] 31 Aug 2019 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. numbering If X and Y are points, then we use the notation XY to denote either the line segment joining X and Y or the length of that line segment, depending on the context. 3. The Orthocenter We start with a known result [4] giving the relationship between the ri when P is the orthocenter. Theorem 3.1. If P is the orthocenter of 4ABC ( [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. r1r3r5 = r2r4r6 The following proof comes from [5] [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: six triangles have same area Proof. If XY Z is a triangle, then [XY Z] will denote the area of that triangle. If two triangles have the same altitude, then the ratio of their areas is the same as the ratio of their bases. Thus [P BD] = [P DC]. Since the centroid divide…
Figure 5
Figure 5. Figure 5: Proof. Let X be the center of incircle 1 ( [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 6
Figure 6. Figure 6: 1 r1 + 1 r3 + 1 r5 = 1 r2 + 1 r4 + 1 r6 Proof. Let R be the circumradius of 4ABC. Let ∠P AB = ∠P BA = α, ∠P BC = ∠P CB = β, and ∠P CA = ∠P AC = γ ( [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: Yellow circles are congruent [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: 1 r1 + 1 r4 + 1 r5 = 1 r2 + 1 r3 + 1 r6 Proof. Let ∠BAD = ∠DAC = a, ∠ABE = ∠EBC = b, and ∠ACF = ∠F CB = c, as shown in [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 9
Figure 9. Figure 9: Note that ∠BP D = a + b. Applying the Law of Sines to 4BP D allows us to compute the values of P B and P D. In the same way, we can compute P C, P E, P A, and P F. We get the following. P A = sin c sin 2b sin(c + a) , P D = sin a sin b sin 2c sin(a + b) sin(a + 2c) , P…
Figure 10
Figure 10. Figure 10: r1r2r3 + r3r4r5 + r3r4r6 = r1r3r4 + r2r3r4 + r4r5r6 Note that there are two independent variables, a and b, and six equations repre￾senting the values of the ri . Thus, variables a and b can be eliminated resulting in an equation relating the ri . Since there are so m…
Figure 11
Figure 11. Figure 11: Since 4a + 4b + 4c = 180◦ , c = 45◦ − a − b. Substitute this value of c into equation (1). Then use the addition formula for cotangent, cot(x + y) = cot x cot y − 1 cot x + cot y , to write all trigonometric expressions in terms of cot a and cot b. In a similar manner…
Figure 12
Figure 12. Figure 12: 5r1 + 6r2 + r4 = r3 + 3r5 + 15r6 [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 13
Figure 13. Figure 13: We then use the Law of Sines to get the following. AF = sin(a + b) sin d sin(a + b + c) , BF = sin(a + b + c + d) sin c sin(a + b + c) , AE = sin(c + d) sin a sin(b + c + d) , CE = sin(a + b + c + d) sin b sin(b + c + d) . Applying the Law of Sines again gives the fol…
Figure 14
Figure 14. Figure 14: 1 r1 + 1 r3 + 1 r4 = 1 r2 + 1 r5 + 1 r6 [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]
Figure 15
Figure 15. Figure 15: 1 r1 + 1 r4 + 1 r6 = 1 r2 + 1 r3 + 1 r5 with the coefficients of the relationship into a database. After all quadruples were examined, I looked at all pairs of entries in the database that had the same set of [PITH_FULL_IMAGE:figures/full_fig_p013_15.png]
Figure 16
Figure 16. Figure 16: K1K3K5 = K2K4K6 Proof. Let ∠BP D = α, ∠DP C = β, and ∠CP E = γ. Noting that vertical angles are equal and using the formula for the area of a triangle in terms of two sides and the sine of the included angle, we have the following six equations. 2K1 = P B · P D · sin …
Figure 17
Figure 17. Figure 17: sets of incircles tangent to circumcircle of 4ABC Open Question 5. Investigate the relationship between the radii in each set of six incircles shown in [PITH_FULL_IMAGE:figures/full_fig_p015_17.png]
Figure 18
Figure 18. Figure 18: sets of incircles tangent to incircle of 4ABC Open Question 6. Investigate the relationship between the radii in each set of six incircles shown in [PITH_FULL_IMAGE:figures/full_fig_p015_18.png]

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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