Treatise of the Quadrilateral

• Treatise of the Quadrilateral is a comprehensive discourse on quadrilaterals, detailing cyclic, secant, and equal-figure approaches across various historical periods.
• It systematically explores methodologies from ancient formulas to modern synthetic, projective, and algebraic techniques, highlighting practical proofs and derivations.
• The treatise connects classical geometric insights with contemporary themes such as quadrilateral tilings, lattice classifications, and configuration space analysis.

Searching arXiv for the cited works to ground the article in current records. arxiv_search: {"query":"id:(Ekhad, 2021) OR id:(Kichenassamy, 4 Jul 2025) OR id:(Papadopoulos, 5 May 2026) OR id:(Beeson, 2020) OR id:(Katzenberger et al., 24 Jan 2026) OR id:(Filippidis et al., 2012) OR id:(Horwitz, 7 May 2026) OR id:(Rabinowitz et al., 5 Jun 2025) OR id:(Berman et al., 29 Mar 2026)", "max_results": 10} “Treatise of the Quadrilateral” appears in distinct mathematical contexts. In one line of usage, it denotes a coherent discourse in which the quadrilateral is the primary geometric object; in another, it is the explicit title of a work, most notably Naṣīr al-Dīn al-Ṭūsī’s treatise on spherical geometry. This suggests a broader designation for a quadrilateral-centered mode of exposition: Brahmagupta’s cyclic quadrilateral theory in the Brāhmasphuṭasiddhānta, Ṭūsī’s secant-figure geometry, Euclidean foundational work on equal quadrilaterals, and modern studies of isogonal conjugation, concurrence, conics, algebraic identities, and configuration spaces all fit that pattern (Kichenassamy, 4 Jul 2025, Papadopoulos, 5 May 2026, Beeson, 2020).

1. Ancient Indian quadrilateral discourse

Brahmagupta’s discussion of cyclic quadrilaterals occurs in chapter XII of the Brāhmasphuṭasiddhānta, in propositions XII.21–32. Kichenassamy’s analysis treats this segment as a coherent discourse, not as a loose list of formulas, and in precisely this sense it functions as a “treatise of the quadrilateral” embedded in a larger astronomical text (Kichenassamy, 4 Jul 2025). Within XII.21–32 Brahmagupta introduces the tricaturbhuja, gives his area formula, states and exploits Pythagorean-type relations, discusses perpendiculars, segments, and portions in cyclic quadrilaterals, states the result later called “Brahmagupta’s theorem” on perpendicular diagonals, and gives expressions for diagonals in terms of sides.

The central metric result is the exact area formula for the cyclic quadrilateral: $$K = \sqrt{(s-a)(s-b)(s-c)(s-d)}, \qquad s = \frac{a+b+c+d}{2}.$$ Brahmagupta also gives a “coarse” area rule,

$$K_{\text{coarse}} = \frac{a+c}{2}\cdot\frac{b+d}{2},$$

which depends on which sides are considered opposite, whereas the refined formula does not. Kichenassamy’s reading insists that the figure is not any quadrilateral, but a cyclic quadrilateral, and that the special term tricaturbhuja is crucial. XII.21 speaks of four sides, while XII.27 asserts that there is a circle passing through the corners; from this, the paper infers that the tricaturbhuja is obtained by adding a fourth vertex on the circumcircle of a given triangle (Kichenassamy, 4 Jul 2025).

A second pillar of the discourse is Brahmagupta’s treatment of perpendicularity without explicit angle language. In the theorem later attributed to Chasles, the relevant vocabulary is not “right angle” but perpendiculars, segments (āvādha), and portions (khanda). Kichenassamy interprets the phrase “segments and portions coincide” as encoding the condition that the foot of a perpendicular coincides with the diagonal intersection, so that orthogonality is expressed operationally rather than through angle measure. This is paired with XII.24, which at first looks like a redundant triple statement of Pythagoras, but is read instead as a mini-derivation spread over three right triangles in a single circular configuration (Kichenassamy, 4 Jul 2025).

The historiographical importance of this material lies in method as much as content. Kichenassamy’s paper argues that unusual expressions such as tricaturbhuja, the order of presentation, and apparent repetitions are structural clues. A plausible implication is that a historically faithful “treatise of the quadrilateral” in this tradition is inseparable from literary form: rules, derivations, and definitions are compressed into verse, and the reader is expected to reconstruct the geometry actively.

2. Al-Ṭūsī and the secant figure

Naṣīr al-Dīn al-Ṭūsī’s famous work on spherical trigonometry is usually known in modern literature as the Treatise of the Quadrilateral, after the title used by Alexandre Carathéodory in 1891. The original Arabic title,

$$\textit{Kashf al-qināʿ ʿan asrār al-shakl al-qaṭṭāʿ},$$

is closer in meaning to “Disclosing the secrets of the secant figure.” Here the “quadrilateral” is the complete (spherical) quadrilateral, also called the sector figure or secant figure, which underlies Menelaus’ theorem (Papadopoulos, 5 May 2026).

In the Euclidean sector figure, with notation $A,G,E,D$ and auxiliary points $B,C$, Menelaus’ theorem is written as

$\frac{AG}{BG} = \frac{AC}{CD}\cdot\frac{DE}{EB}.$

On the sphere, the corresponding theorem becomes

$$\frac{\sin AG}{\sin BG} = \frac{\sin AC}{\sin CD}\cdot\frac{\sin DE}{\sin EB}.$$

This theorem, used by Ptolemy in the Almagest, was the principal computational tool of spherical astronomy before the full development of spherical trigonometry. Ṭūsī’s treatise contains “much more than Menelaus’ theorem,” namely “a complete system of spherical trigonometric formulae, with complete proofs” (Papadopoulos, 5 May 2026).

The work is divided into five books: on compounded ratios; on the plane quadrilateral figure; on lemmas for the spherical quadrilateral figure; on the spherical quadrilateral figure and the ratios found therein; and, finally, on “the methods that dispense with the theory of the quadrilateral figure, for what concerns the knowledge of the arcs of great circles.” Book V contains the spherical law of sines,

$$\frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C},$$

as well as cosine and tangent formulas, the polar triangle, the supplementary figure, and the shadowed figure. In Ṭūsī’s terminology, “the shadow is what we call the tangent of an angle” (Papadopoulos, 5 May 2026).

Historically, the treatise is a hinge text. It is at once the last major exposition of secant-figure geometry and a systematic presentation of the methods that replace it in practice. Papadopoulos emphasizes that Book V “dispenses with” the quadrilateral computationally while still preserving it conceptually as the starting point. In that respect, the title “Treatise of the Quadrilateral” is exact even where the mature trigonometric system has outgrown the original figure.

3. Euclidean foundations: equal quadrilaterals and Saccheri-type quadrilaterals

Euclid uses an undefined notion of “equal figures,” applying to triangles and quadrilaterals the common notions about equals added to equals or subtracted from equals. Beeson and collaborators replace this primitive usage by definitions built from congruence, parallelism, and an early theory of proportions. Their definitions of equal rectangles, equal triangles, and equal quadrilaterals are designed to recover the exact properties Euclid needs, while removing the need for additional axioms (Beeson, 2020).

The construction proceeds by using rectangles as “area carriers.” Equal rectangles are defined by a configuration condition equivalent to a proportionality relation between side lengths. Equal triangles are then defined as triangles whose first circumscribed rectangles are equal rectangles. For quadrilaterals, a circumscribed rectangle is chosen with sides parallel to a diagonal and containing all four vertices on its sides, and two quadrilaterals are equal quadrilaterals if some circumscribed rectangle of one is equal rectangle to some circumscribed rectangle of the other. Within this framework, Euclid’s propositions about parallelograms on the same base and in the same parallels, complements, cutting and pasting, and the construction of parallelograms equal to given figures all become theorems rather than axioms (Beeson, 2020).

A different foundational use of the quadrilateral appears in the study of the parallel postulate. The paper on “the acute angles hypothesis on a two right-angled isosceles quadrilateral, with a rectilinear summit side” treats a Saccheri-type configuration inside a Hilbert plane. It stresses the model-theoretic viewpoint: consistency is existence of a model, and independence is exhibited by models in which a statement holds and models in which it fails. In that setting, Saccheri-type quadrilaterals become probes of the status of Euclid’s fifth postulate (Filippidis et al., 2012).

The paper’s culminating statement is that “the type of the summit side angles (acute, obtuse, right) of a Saccheri quadrilateral depends on the curvature of the summit side” (Filippidis et al., 2012). This ties the classical right/acute/obtuse trichotomy to the later language of Euclidean, hyperbolic, and positively curved models. Taken together, these foundational strands show that a treatise of the quadrilateral can be either metric and constructive, as in Euclid, or logical and model-theoretic, as in the afterlife of Saccheri.

4. Modern Euclidean and projective traditions

In modern Euclidean geometry, one quadrilateral-centered theorem singled out explicitly as belonging to a broader “treatise” is Leversha’s “quartet of isogonal conjugates.” For a non-cyclic quadrilateral $ABCD$, let $A^*$ be the isogonal conjugate of $$K_{\text{coarse}} = \frac{a+c}{2}\cdot\frac{b+d}{2},$$0 with respect to $$K_{\text{coarse}} = \frac{a+c}{2}\cdot\frac{b+d}{2},$$1, and similarly for $$K_{\text{coarse}} = \frac{a+c}{2}\cdot\frac{b+d}{2},$$2. Then each original vertex becomes the circumcenter of the quadrilateral formed by the other three conjugates together with itself. Equivalently,

$$K_{\text{coarse}} = \frac{a+c}{2}\cdot\frac{b+d}{2},$$3

Ekhad’s note gives a one-line Maple verification of this theorem and remarks that Leversha’s “quartet of isogonal conjugates” fits naturally into a more general “treatise” on quadrilaterals (Ekhad, 2021).

A second modern strand is synthetic-projective. In a convex quadrilateral $$K_{\text{coarse}} = \frac{a+c}{2}\cdot\frac{b+d}{2},$$4, with points $$K_{\text{coarse}} = \frac{a+c}{2}\cdot\frac{b+d}{2},$$5, $$K_{\text{coarse}} = \frac{a+c}{2}\cdot\frac{b+d}{2},$$6, $$K_{\text{coarse}} = \frac{a+c}{2}\cdot\frac{b+d}{2},$$7, $$K_{\text{coarse}} = \frac{a+c}{2}\cdot\frac{b+d}{2},$$8, the single global ratio

$$K_{\text{coarse}} = \frac{a+c}{2}\cdot\frac{b+d}{2},$$9

controls a large concurrency theory. Under the restrictive condition $$\textit{Kashf al-qināʿ ʿan asrār al-shakl al-qaṭṭāʿ},$$0, seven lines are concurrent: $$\textit{Kashf al-qināʿ ʿan asrār al-shakl al-qaṭṭāʿ},$$1 Without that restriction, four derived points $$\textit{Kashf al-qināʿ ʿan asrār al-shakl al-qaṭṭāʿ},$$2 still organize the configuration, and the paper develops an extension of a transversal theorem from triangles to quadrilaterals (Cozma, 2012). This is a genuinely quadrilateral analogue of Ceva–Menelaus-type triangle geometry.

A third strand is explicitly projective and uses quadrilateral tilings. In the manifold-based theory of projective incidence theorems, the basic tile is a quadrangle whose coherency condition is

$$\textit{Kashf al-qināʿ ʿan asrār al-shakl al-qaṭṭāʿ},$$3

Quadrilateral tilings on closed orientable 2-manifolds are shown to be equivalent in strength to proofs using exclusively Menelaus configurations, and to be at most as strong as proofs using bi-quadratic final polynomials (Katzenberger et al., 24 Jan 2026). Here the quadrilateral is not merely an object of study but the local atom of a proof system.

The projective tradition also contains a cautionary chapter. In the modern analysis of Chasles’ Quadrilateral Theorem, several classical and modern formulations are shown to be affected by subtle ambiguities caused by multiple solutions in the underlying constructions. Correct formulations are then given in a projective framework, and the underlying incidence structure is explained by the Cayley–Bacharach theorem (Berman et al., 29 Mar 2026). This suggests that a modern treatise of the quadrilateral must include not only elegant statements but also a careful account of hidden multiplicities.

5. Algebraic, conic, and derived quadrilateral structures

A major algebraic theme is Euler’s quadrilateral identity. In its classical Euclidean form, if $$\textit{Kashf al-qināʿ ʿan asrār al-shakl al-qaṭṭāʿ},$$4 and $$\textit{Kashf al-qināʿ ʿan asrār al-shakl al-qaṭṭāʿ},$$5 are the midpoints of the diagonals $$\textit{Kashf al-qināʿ ʿan asrār al-shakl al-qaṭṭāʿ},$$6 and $$\textit{Kashf al-qināʿ ʿan asrār al-shakl al-qaṭṭāʿ},$$7 of a quadrilateral $$\textit{Kashf al-qināʿ ʿan asrār al-shakl al-qaṭṭāʿ},$$8, then

$$\textit{Kashf al-qināʿ ʿan asrār al-shakl al-qaṭṭāʿ},$$9

The recent vector treatment derives this from the parallelogram identity via Apollonius’ identity and then generalizes it to $A,G,E,D$0 vectors in an arbitrary real or complex inner product space (Murad, 8 Mar 2026). In this setting, Euler’s theorem is one case of a general identity involving sums of squared norms.

A conic counterpart is given by Besant quadrilaterals. If an ellipse is inscribed in a convex quadrilateral $A,G,E,D$1 so that one focus is equidistant from the four vertices, that focus is necessarily the circumcenter $A,G,E,D$2 of $A,G,E,D$3, and the other distinguished point is the diagonal intersection $A,G,E,D$4. Horwitz proves that such an inscribed ellipse exists if and only if $A,G,E,D$5 is orthodiagonal. More precisely, for a cyclic quadrilateral that is not a parallelogram, if one focus of an inscribed ellipse equals $A,G,E,D$6 or $A,G,E,D$7, then the other focus equals the other point; conversely, if $A,G,E,D$8 is also orthodiagonal, there exists an inscribed ellipse whose foci are exactly $A,G,E,D$9 and $B,C$0 (Horwitz, 7 May 2026).

Derived quadrilaterals formed from triangle centers constitute another modern chapter. In one construction, the diagonals of a quadrilateral form four associated triangles, called half triangles; choosing the same triangle center in each produces a central quadrilateral (Rabinowitz et al., 5 Jun 2025). In an earlier version, an interior point $B,C$1 divides a convex quadrilateral into four radial triangles, and again a chosen triangle center in each triangle defines a central quadrilateral (Rabinowitz et al., 2022). Rabinowitz and Suppa show, for example, that when the center is $B,C$2 (the centroid of a half triangle), the central quadrilateral is similar to the reference quadrilateral with ratio $B,C$3, so that

$B,C$4

For $B,C$5, they obtain equal area, and for $B,C$6, identities involving the quadrilateral’s Poncelet point and centroid (Rabinowitz et al., 5 Jun 2025).

These algebraic and conic developments broaden the scope of a treatise of the quadrilateral beyond side-angle theorems. The quadrilateral becomes a site where vector identities, inscribed conics, and functorial constructions from triangle centers all coexist.

6. Arithmetic and configuration-space extensions

Arithmetic quadrilateral theory appears prominently in the study of lattice equable quadrilaterals, quadrilaterals in $B,C$7 whose area equals their perimeter. In the tangential and extangential cases, the classification is unexpectedly rigid. Up to Euclidean motions, there are only 6 convex tangential lattice equable quadrilaterals, while the concave tangential ones are arranged in 7 infinite families. Apart from the kites, up to Euclidean motions there is only one concave extangential lattice equable quadrilateral, while there are infinitely many convex ones (Aebi et al., 2021). Here the treatise of the quadrilateral becomes a meeting point of Euclidean geometry and quadratic Diophantine equations.

At the opposite conceptual extreme lies quadrilateralland. In relational particle mechanics, the pure-shape configuration space of 4 point particles in 2D, after factoring out overall translations, rotations, and scale, is

$B,C$8

Quadrilateralland therefore interprets $B,C$9 as the space of relational quadrilateral shapes. Its isometry group is $\frac{AG}{BG} = \frac{AC}{CD}\cdot\frac{DE}{EB}.$0, which yields $\frac{AG}{BG} = \frac{AC}{CD}\cdot\frac{DE}{EB}.$1-like conserved quantities. Anderson identifies analogues of isospin and hypercharge, with

$\frac{AG}{BG} = \frac{AC}{CD}\cdot\frac{DE}{EB}.$2

and gives explicit Kuchař observables: in the pure-shape model, they are precisely the functions of the shape variables and shape momenta (Anderson, 2012).

This suggests a final enlargement of the expression. A “Treatise of the Quadrilateral” no longer names only a chapter of classical geometry or a single medieval book. It can also designate a family of mathematical programs in which the quadrilateral organizes area theory, spherical trigonometry, projective incidence, algebraic identities, arithmetic classification, and even the configuration space of a relational universe.