Besant quadrilaterals
Pith reviewed 2026-05-21 08:45 UTC · model grok-4.3
The pith
A quadrilateral admits an inscribed ellipse with one focus equidistant from the vertices if and only if its diagonals are perpendicular, placing the other focus at their intersection.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
If an ellipse is inscribed in a quadrilateral Q so that one focus is equidistant from the four vertices, the other focus must be at the intersection of the diagonals. The paper proves the converse and shows that such an inscribed ellipse exists if and only if Q is orthodiagonal. Additional results are given when Q is a trapezoid.
What carries the argument
The formula for the coefficients of an ellipse inscribed in a quadrilateral, which locates the foci and relates the equidistant point from the vertices to the intersection of the diagonals.
If this is right
- Orthodiagonal quadrilaterals always admit an inscribed ellipse with one focus at the equidistant point from the vertices and the other at the diagonal intersection.
- The result includes specific additional properties when the quadrilateral is a trapezoid.
- The existence condition is both necessary and sufficient for the described ellipse configuration.
Where Pith is reading between the lines
- The characterization might serve as a geometric test for whether a quadrilateral has perpendicular diagonals.
- Similar focus conditions could be explored for other conics inscribed in quadrilaterals or polygons.
- The connection between the equidistant point and diagonal intersection may extend to applications in geometry construction.
Load-bearing premise
The formula for the coefficients of an ellipse inscribed in a quadrilateral correctly determines the locations of the foci for quadrilaterals that admit an inscribed ellipse.
What would settle it
A non-orthodiagonal quadrilateral that still admits an inscribed ellipse with one focus equidistant from all four vertices would falsify the if-and-only-if claim.
read the original abstract
We solve the following problem of W.H. Besant using a formula for the coefficients of an ellipse inscribed in a quadrilateral, $Q$: \enquote{If an ellipse be inscribed in a quadrilateral so that one focus is equidistant from the four vertices(call that point $EP$), the other focus must be at the intersection of the diagonals(call that point $IP$).} We also prove somewhat more than just solving Besant's problem itself, though it would be nice to see the details of the geometric approach proposed by Besant. More precisely, we also prove the converse result and additional results when $Q$ is a trapezoid. Finally, we show that such an inscribed ellipse exists if and only if $Q$ is orthodiagonal.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper solves W.H. Besant's problem for a quadrilateral Q with an inscribed ellipse having one focus equidistant from the four vertices (denoted EP): it shows that the other focus must lie at the intersection of the diagonals (IP). Using a formula for the coefficients of an inscribed ellipse, the authors derive focus locations, establish the orthodiagonal criterion, prove the converse, treat the trapezoid case separately, and conclude that such an inscribed ellipse exists if and only if Q is orthodiagonal.
Significance. If the derivations hold and the existence statement is corrected to account for the tangential condition, the work would furnish an algebraic treatment of a classical geometric problem, clarifying the interplay between focus equidistance, diagonal intersection, and orthodiagonality for tangential quadrilaterals. The use of an explicit coefficient formula for the inscribed ellipse is a concrete strength that could support reproducibility if the intermediate algebraic steps are fully expanded.
major comments (2)
- [Abstract] Abstract (final sentence) and the corresponding existence claim in the main text: the statement that 'such an inscribed ellipse exists if and only if Q is orthodiagonal' is incorrect. Any ellipse inscribed in Q (tangent to all four sides) requires Q to be tangential, i.e., a + c = b + d by Pitot's theorem. Orthodiagonality is independent of this condition; the kite with side lengths 3, 4, 3, 5 is orthodiagonal (perpendicular diagonals) yet fails a + c = b + d and therefore admits no inscribed ellipse at all. This directly falsifies the 'if' direction of the claimed equivalence and is load-bearing for the paper's final theorem.
- [Main derivation section] The derivation of focus locations from the coefficient formula (invoked to reach the orthodiagonal criterion) is not fully expanded in the provided text. Without the intermediate algebraic steps that convert the equidistance condition at EP into the requirement that the second focus lies at the diagonal intersection, it is impossible to verify whether hidden assumptions (e.g., the quadrilateral already being tangential) are tacitly used.
minor comments (1)
- [Introduction] Notation for EP and IP is introduced in the abstract but should be restated with explicit definitions at the first use in the body text for clarity.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript and for identifying these important points. We address each major comment below and agree that revisions are needed to correct and clarify the presentation.
read point-by-point responses
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Referee: [Abstract] Abstract (final sentence) and the corresponding existence claim in the main text: the statement that 'such an inscribed ellipse exists if and only if Q is orthodiagonal' is incorrect. Any ellipse inscribed in Q (tangent to all four sides) requires Q to be tangential, i.e., a + c = b + d by Pitot's theorem. Orthodiagonality is independent of this condition; the kite with side lengths 3, 4, 3, 5 is orthodiagonal (perpendicular diagonals) yet fails a + c = b + d and therefore admits no inscribed ellipse at all. This directly falsifies the 'if' direction of the claimed equivalence and is load-bearing for the paper's final theorem.
Authors: We agree with this assessment. The final existence claim in the abstract and main text is stated too broadly and does not account for the tangential condition (a + c = b + d) required by Pitot's theorem for any inscribed ellipse to exist. The provided kite example correctly demonstrates that orthodiagonality alone is insufficient. In the revised manuscript we will correct the statement to read that such an inscribed ellipse exists if and only if Q is both tangential and orthodiagonal. We will also add a brief reminder of Pitot's theorem at the relevant points to make the necessary conditions explicit. revision: yes
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Referee: [Main derivation section] The derivation of focus locations from the coefficient formula (invoked to reach the orthodiagonal criterion) is not fully expanded in the provided text. Without the intermediate algebraic steps that convert the equidistance condition at EP into the requirement that the second focus lies at the diagonal intersection, it is impossible to verify whether hidden assumptions (e.g., the quadrilateral already being tangential) are tacitly used.
Authors: We acknowledge that the algebraic steps deriving the focus locations from the coefficient formula are presented in condensed form. In the revision we will insert the missing intermediate calculations, explicitly showing the transition from the equidistance condition at EP to the conclusion that the second focus must lie at the diagonal intersection. These expanded steps will also clarify that the tangential condition is presupposed (or separately verified) before applying the coefficient formula, thereby removing any ambiguity about hidden assumptions. revision: yes
Circularity Check
No circularity; derivation uses external formula for ellipse coefficients
full rationale
The paper invokes a pre-existing formula for the coefficients of an ellipse inscribed in a quadrilateral to locate foci and establish the claimed equivalence with orthodiagonal quadrilaterals. This formula is treated as an independent input rather than derived from or defined in terms of the target result (one focus equidistant from vertices implying the other at diagonal intersection). No steps reduce by construction to fitted parameters, self-citations, or ansatzes smuggled from prior work by the same author; the central claims are obtained by algebraic manipulation of the given formula and geometric properties without self-referential forcing. The derivation remains self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard algebraic and geometric properties of ellipses inscribed in quadrilaterals hold, including the relation between coefficients and focus locations.
Reference graph
Works this paper leans on
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[1]
C. Bond, ”A New Algorithm for Scan Conversion of a General Ellipse”, preprint, http://www.crbond.com/papers/ell alg.pdf
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[2]
W.H. Besant, Conic Sections Treated Geometrically, George Bell and Sons Educational Catalogue, Project Gutenberg ebook number 29913, 2009
work page 2009
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[3]
W.H. Besant, Solutions of Examples in Conic Sections Treated Geometrically, 3rd edition, revised, Cambridge, 1890
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[4]
G. D. Chakerian, A Distorted View of Geometry, MAA, Mathematical Plums, Washington, DC, 1979, 130-150
work page 1979
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[5]
Alan Horwitz, Ellipses Inscribed in, and Circumscribed about, Quadrilaterals, Chapman and Hall, 2024, ISBN 9781032622590, 149 pages
work page 2024
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[6]
Alan Horwitz, ”Ellipses of maximal area and of minimal eccentricity inscribed in a convex quadri- lateral”, Australian Journal of Mathematical Analysis and Applications, Volume 2, Issue 1 (2005), 1-12
work page 2005
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[7]
Alan Horwitz, Dynamics of ellipses inscribed in triangles, Journal of Science, Environment and Technology, Volume 5, Issue 1 (2016), 1-21
work page 2016
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[8]
Morris Marden, ”A note on the zeros of the sections of a partial fraction, Bulletin of the AMS 51 (1945), 935-940
work page 1945
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[9]
Mohamed Ali Said, ”Calibration of an Ellipse’s Algebraic Equation and Direct Determination of its Parameters”, Acta Mathematica Academiae Paedagogicae Ny regyh aziensis Vol.19, No. 2 (2003), 221-225
work page 2003
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[10]
https://en.m.wikipedia.org/wiki/Conic section
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[11]
https://mathworld.wolfram.com/Ellipse.html 13
discussion (0)
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