Fact-checked by Grok 2 weeks ago

Ptolemy's theorem

Ptolemy's theorem states that in a cyclic quadrilateral—a quadrilateral whose vertices all lie on a single circle—the product of the lengths of the diagonals is equal to the sum of the products of the lengths of the opposite sides.^[1] For a quadrilateral ABCD inscribed in a circle, this relation is expressed as AC \times BD = AB \times CD + AD \times BC.^[2] Named after the Greco-Roman mathematician and astronomer Claudius Ptolemy (c. 100–170 CE), the theorem appears in his influential astronomical treatise Almagest, where it served as a key tool for constructing tables of chords in a circle, effectively laying groundwork for early trigonometry.^[3] Ptolemy provided a proof using geometric constructions involving similar triangles, though modern analyses note a small gap in his argumentation that was later filled by subsequent mathematicians.^[3] The theorem's significance extends beyond its historical context; it generalizes the Pythagorean theorem, which emerges as a special case when the cyclic quadrilateral degenerates into a rectangle.^[1] Additionally, combined with the law of sines, it yields the trigonometric addition formulas for sine and cosine, underscoring its foundational role in plane geometry and its applications in astronomy, navigation, and computational methods.^[2] The converse of Ptolemy's theorem also holds: if a quadrilateral satisfies the side-diagonal relation, then it must be cyclic, providing a criterion for determining whether four points lie on a common circle.^[4]

Statement and Illustration

Formal Statement

A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a single circle, allowing it to be inscribed in that circle. Ptolemy's theorem asserts that if ABCD is a cyclic quadrilateral, with consecutive sides of lengths AB, BC, CD, and DA, and diagonals of lengths AC and BD, then the product of the diagonals equals the sum of the products of the opposite sides:

AC \cdot BD = AB \cdot CD + AD \cdot BC

^[1]^[5] In conventional notation, the sides are often labeled as a = AB, b = BC, c = CD, d = DA, while the diagonals are p = AC and q = BD, yielding the relation p q = ac + bd.^[2] This equality characterizes cyclic quadrilaterals, as for non-cyclic quadrilaterals the corresponding relation is a strict inequality.

Geometric Illustration

Ptolemy's theorem applies to a cyclic quadrilateral, which is a four-sided figure with all vertices lying on the circumference of a single circle. Consider a labeled diagram of quadrilateral ABCD inscribed in a circle, where points A, B, C, and D are marked sequentially around the circle. The sides are denoted as AB, BC, CD, and DA, while the diagonals are AC and BD, intersecting at some point inside the quadrilateral. This inscription ensures that opposite angles sum to 180 degrees, creating the geometric configuration necessary for the theorem's equality to hold.^[5] To illustrate the theorem in practice, take a specific example of a rectangle ABCD, which is a special case of a cyclic quadrilateral. Assign lengths AB = 3, BC = 4, CD = 3, and DA = 4; the diagonals AC and BD each measure 5, as determined by the Pythagorean theorem applied to the right triangles formed. Applying the theorem, the product of the diagonals is 5 × 5 = 25, while the sum of the products of opposite sides is (3 × 3) + (4 × 4) = 9 + 16 = 25, verifying the equality.^[6] The inscription in the circle is essential because it imposes the cyclic condition that makes the equality true; for non-cyclic quadrilaterals, the relationship becomes an inequality instead. Intuitively, the theorem reveals a balance in the cyclic quadrilateral's geometry, where the product of the diagonals exactly matches the sum of the products of the opposite sides, reflecting the harmonious constraints of the enclosing circle.^[2]

Historical Background

Ptolemy's Contribution in the Almagest

Claudius Ptolemy (c. 100–170 AD), an Alexandrian mathematician and astronomer, authored the Almagest, a comprehensive treatise on astronomy that synthesized and advanced Greek mathematical traditions.^[7] This work, formally titled Mathematical Syntaxis, established a geocentric model of the universe and introduced rigorous computational methods for celestial phenomena.^[8] In the Almagest, Ptolemy utilized a geometric relation for cyclic quadrilaterals—now recognized as Ptolemy's theorem—to construct a table of chords, which functioned as an early trigonometric tool for astronomical calculations.^[9] These chords represented straight-line distances subtended by arcs in a circle, enabling the determination of angular separations between stars, planets, and other celestial objects.^[10] By basing the table on a circle of radius 60 parts, Ptolemy ensured compatibility with sexagesimal arithmetic, a system that facilitated precise divisions of angles and arcs up to half-degree increments.^[11] Ptolemy specifically applied the theorem to derive chord lengths for use in his planetary models, which accounted for observed motions such as retrogradations and eccentric orbits.^[9] For instance, the relation allowed him to compute unknown chords from combinations of known ones, supporting predictions of planetary positions relative to the fixed stars and the calculation of eclipse timings.^[10] The first known explicit statement of the theorem occurs in Book I, Chapter 10 of the Almagest, where Ptolemy articulates that, for a quadrilateral inscribed in a circle, the product of the lengths of the diagonals equals the sum of the products of the lengths of the opposite sides.^[11] Ptolemy provided a proof of the theorem using geometric constructions involving similar triangles, though it contains a slight gap that was later identified and addressed by historians such as Sir Thomas Heath.^[3] This formulation provided a foundational method for extending the chord table beyond basic angles, enhancing the accuracy of his astronomical framework.^[11]

Earlier Origins and Later Developments

The roots of Ptolemy's theorem trace back to ancient Greek geometry, where properties of cyclic quadrilaterals were explored in Euclid's Elements around 300 BCE, including results in Book III, Proposition 21, that implicitly relate sides and angles in inscribed figures, laying groundwork for later developments in chord calculations.^[3] Although no explicit statement of the theorem appears in surviving pre-Ptolemaic texts, earlier Hellenistic astronomers like Hipparchus (c. 190–120 BCE) and Menelaus (c. 70–130 CE) constructed chord tables for spherical trigonometry, potentially employing equivalent relations without formal proof, as suggested by analyses of their methods.^[3] These contributions indicate that the theorem's core ideas may have circulated orally or implicitly within Hellenistic mathematics, though definitive attribution remains elusive due to fragmentary records.^[3] During the medieval period, the Almagest—containing Ptolemy's formulation—was translated into Arabic multiple times in the ninth century, including versions by al-Ḥajjāj ibn Yūsuf ibn Maṭar, Isḥāq ibn Hunayn, and Thābit ibn Qurra, preserving and disseminating the work across the Islamic world.^[12] These translations facilitated engagement by Islamic scholars, notably Naṣīr al-Dīn al-Ṭūsī (1201–1274), who in his redaction and commentary on the Almagest (completed around 1247) replaced Ptolemy's geometric chord theorems, including the one for cyclic quadrilaterals, with trigonometric equivalents to align with contemporary methods.^[13] By the twelfth century, Latin translations, such as Gerard of Cremona's from Arabic around 1175, introduced the theorem to Europe, bridging Greek and Renaissance mathematics.^[14] In the nineteenth and twentieth centuries, Ptolemy's theorem experienced a revival in Euclidean geometry education, appearing in influential texts like Problems and Solutions in Euclidean Geometry by Ali A. Aref and William Wernick (1968), which integrated it into problem-solving curricula.^[15] It gained prominence in mathematical competitions, featuring in Olympiad problems from the mid-twentieth century onward, as highlighted in resources like Evan Chen's Euclidean Geometry in Mathematical Olympiads (2016), where it serves as a key tool for tackling cyclic quadrilateral configurations.^[16] This resurgence underscored its enduring utility in pure geometry, distinct from its astronomical origins.

Proofs

Visual Proof

One notable visual proof of Ptolemy's theorem employs a dissection technique with scaled triangles to intuitively reveal the equality without algebraic steps. This proof, originally presented as a "proof without words," constructs and rearranges dilated versions of triangles from the cyclic quadrilateral to form congruent figures representing both sides of the equation. The diagram begins with cyclic quadrilateral ABCD, where diagonals AC and BD intersect at point O, dividing the figure into four triangles: AOB, BOC, COD, and DOA. From this setup, three key triangles are selected for scaling: △ABC (spanning sides AB, BC and diagonal AC), △ABD (spanning sides AB, AD and diagonal BD), and △BCD (spanning sides BC, CD and diagonal BD). These triangles collectively encompass the quadrilateral's structure and highlight the side and diagonal relationships. The construction proceeds by dilating each triangle by a specific linear factor corresponding to an opposite or adjacent length, preserving angles and enabling visual matching of segments: △BCD is dilated by the factor AB, △ABD by the factor BC, and △ABC by the factor BD. The resulting scaled figures have enlarged sides that embody the products AB \cdot CD, AD \cdot BC, and related terms, with the scalings chosen to align corresponding elements geometrically. These dilated triangles are then dissected into pieces and reassembled in two distinct configurations. In the first configuration, the pieces form a parallelogram (or equivalent congruent shape) with dimensions directly tied to the diagonals AC and BD, visually representing the product AC \cdot BD through the overall length and width. In the second configuration, the same pieces are rearranged to form adjacent regions corresponding to the products AB \cdot CD and AD \cdot BC, where the segments align additively to match the length of the first figure's corresponding dimension. The congruence of these figures—evident from matching boundaries, angles, and enclosed areas—establishes the equality AC \cdot BD = AB \cdot CD + AD \cdot BC without measurement. The key insight lies in the overlapping and adjacent regions created during rearrangement, where the scaled segments for AB \cdot CD and AD \cdot BC visually "add up" to fill the space equivalent to AC \cdot BD, emphasizing the theorem's geometric balance through spatial alignment rather than computation. This method's primary advantage is its accessibility, offering an intuitive grasp of the theorem for learners focused on spatial reasoning and diagram interpretation, independent of coordinate systems or equations.

Proof by Similarity of Triangles

To prove Ptolemy's theorem using similarity of triangles, consider a cyclic quadrilateral ABCD. Draw diagonal AC. Select a point M on diagonal BD such that ∠ACB = ∠MCD.^[2] Triangles ABC and DMC are similar because ∠BAC = ∠BDC (both inscribed angles subtending arc BC) and ∠ACB = ∠MCD by construction, making the third angles equal by the sum of angles in a triangle.^[2] From this similarity, the corresponding sides are proportional, yielding

\frac{CD}{MD} = \frac{AC}{AB},

AB \cdot CD = AC \cdot MD.

^[2] ∠BCM = ∠ACD because ∠BCD = ∠BCM + ∠MCD = ∠BCA + ∠ACD and ∠MCD = ∠BCA by construction. Additionally, ∠CBM = ∠CAD (both inscribed angles subtending arc CD). Thus, triangles BCM and ACD are similar by AA similarity.^[2] The corresponding sides give

\frac{BC}{BM} = \frac{AC}{AD},

BC \cdot AD = AC \cdot BM.

^[2] Adding the two equations

AB \cdot CD + BC \cdot AD = AC \cdot MD + AC \cdot BM = AC \cdot (BM + MD) = AC \cdot BD,

which is Ptolemy's theorem.^[2]

Proof by Trigonometric Identities

One approach to proving Ptolemy's theorem employs the law of cosines in the triangles formed by each diagonal of the cyclic quadrilateral, exploiting the property that opposite angles sum to 180° to relate the cosine terms.^[17] Consider cyclic quadrilateral ABCD with sides AB = a, BC = b, CD = c, DA = d, and diagonals AC = p, BD = q. The theorem states that pq = ac + bd.^[17] First, apply the law of cosines to diagonal p = AC in triangles ABC and ADC. Let β = ∠ABC. Then,

p^2 = a^2 + b^2 - 2ab \cos \beta

In triangle ADC, ∠ADC = 180° - β, so \cos(180^\circ - \beta) = -\cos \beta. Thus,

p^2 = d^2 + c^2 - 2dc \cos(180^\circ - \beta) = d^2 + c^2 + 2dc \cos \beta.

Equating the expressions for p^2,

a^2 + b^2 - 2ab \cos \beta = d^2 + c^2 + 2dc \cos \beta,

which rearranges to

a^2 + b^2 - c^2 - d^2 = 2(ab + dc) \cos \beta.

Solving for \cos \beta,

\cos \beta = \frac{a^2 + b^2 - c^2 - d^2}{2(ab + dc)}.

Substitute this into the first expression for p^2:

p^2 = a^2 + b^2 - 2ab \left( \frac{a^2 + b^2 - c^2 - d^2}{2(ab + dc)} \right) = a^2 + b^2 - \frac{ab(a^2 + b^2 - c^2 - d^2)}{ab + dc}.

Clearing the denominator yields

p^2 = \frac{(a^2 + b^2)(ab + dc) - ab(a^2 + b^2 - c^2 - d^2)}{ab + dc} = \frac{ab(c^2 + d^2) + dc(a^2 + b^2)}{ab + dc}.

Now repeat the process for diagonal q = BD in triangles ABD and BCD. Let γ = ∠DAB. Then,

q^2 = a^2 + d^2 - 2ad \cos \gamma

In triangle BCD, ∠BCD = 180° - γ, so

q^2 = b^2 + c^2 + 2bc \cos \gamma.

Equating and solving similarly gives

\cos \gamma = \frac{a^2 + d^2 - b^2 - c^2}{2(ad + bc)},

and substituting yields

q^2 = \frac{ad(b^2 + c^2) + bc(a^2 + d^2)}{ad + bc}.

To obtain pq, multiply the expressions for p^2 and q^2:

p^2 q^2 = \left[ \frac{ab(c^2 + d^2) + dc(a^2 + b^2)}{ab + dc} \right] \left[ \frac{ad(b^2 + c^2) + bc(a^2 + d^2)}{ad + bc} \right].

Algebraic expansion of the numerator shows it equals (ac + bd)^2 (ab + dc)(ad + bc), so

p^2 q^2 = (ac + bd)^2,

and since lengths are positive, pq = ac + bd.^[17]

Proof by Inversion

One approach to proving Ptolemy's theorem utilizes circle inversion, a transformation that maps points with respect to a circle of radius k centered at a point O, where the image P' of a point P satisfies |OP| \cdot |OP'| = k^2 and lies on the ray from O through P.^[18] Consider a cyclic quadrilateral ABCD inscribed in a circle \Gamma. Choose the center of inversion O at vertex A and radius k sufficiently large (greater than the diameter of \Gamma). Since \Gamma passes through the center A, its image under inversion is a straight line \ell not passing through A. The vertices B, C, and D map to points B', C', and D' on \ell, preserving the cyclic configuration in the transformed plane.^[18] The key distance relation under inversion states that for distinct points P and Q (neither at the center), the distance between their images is |P'Q'| = k^2 \cdot |PQ| / (|OP| \cdot |OQ|). Applying this to the transformed points on \ell, assuming the order B', D', C' along the line (corresponding to the convex quadrilateral configuration), yields collinearity implying |B'D'| + |D'C'| = |B'C'|. Substituting the distance formula gives:

\frac{k^2 \cdot |BD|}{|AB| \cdot |AD|} + \frac{k^2 \cdot |DC|}{|AD| \cdot |AC|} = \frac{k^2 \cdot |BC|}{|AB| \cdot |AC|}.

Dividing through by k^2 and multiplying both sides by |AB| \cdot |AD| \cdot |AC| simplifies to:

|AC| \cdot |BD| + |AB| \cdot |DC| = |AD| \cdot |BC|.

This is precisely Ptolemy's theorem, relating the products of the opposite sides and diagonals. The inversion preserves angles, ensuring the geometric relations hold, and the equality follows directly from the linear configuration in the image.^[18] In the original quadrilateral, the inverse transformation reverts the equality, confirming it for the cyclic case. If ABCD were not cyclic, the images B', C', D' would not be collinear, leading to a strict inequality via the triangle inequality in the transformed figure.^[18]

Proof Using Complex Numbers

One approach to proving Ptolemy's theorem employs the complex plane, where the vertices of the cyclic quadrilateral ABCD are represented by complex numbers z_1, z_2, z_3, and z_4 respectively. The distances between points correspond to the magnitudes of their differences: AB = |z_2 - z_1|, BC = |z_3 - z_2|, CD = |z_4 - z_3|, DA = |z_1 - z_4|, AC = |z_3 - z_1|, and BD = |z_4 - z_2|. Since the points lie on a circle, they are concyclic, which is equivalent to their cross-ratio being a real number.^[19] Consider the following expressions derived from permutations of the cross-ratio:

r_1 = \frac{(z_2 - z_1)(z_3 - z_4)}{(z_2 - z_4)(z_3 - z_1)}, \quad r_2 = \frac{(z_2 - z_3)(z_1 - z_4)}{(z_2 - z_4)(z_1 - z_3)}.

These are cross-ratios (or closely related anharmonic ratios) for the points in different orders. A key property of cross-ratios is that r_1 + r_2 = 1 holds algebraically for any complex numbers z_1, z_2, z_3, z_4.^[19] Because the points are concyclic, both r_1 and r_2 are real numbers; for a convex quadrilateral in the standard order around the circle, they are positive.^[19] Taking absolute values yields |r_1| + |r_2| = 1. Substituting the magnitudes gives

\left| r_1 \right| = \frac{|z_2 - z_1| \cdot |z_3 - z_4|}{|z_2 - z_4| \cdot |z_3 - z_1|}, \quad \left| r_2 \right| = \frac{|z_2 - z_3| \cdot |z_1 - z_4|}{|z_2 - z_4| \cdot |z_1 - z_3|}.

Note that |z_3 - z_1| = |z_1 - z_3|, so the denominators are identical. Thus,

\frac{|z_2 - z_1| \cdot |z_3 - z_4| + |z_2 - z_3| \cdot |z_1 - z_4|}{|z_2 - z_4| \cdot |z_1 - z_3|} = 1,

which rearranges to

AB \cdot CD + AD \cdot BC = AC \cdot BD.

This establishes Ptolemy's theorem. The proof relies on the algebraic invariance of the cross-ratio sum and the geometric condition that concyclicity makes the ratios real and positive, ensuring equality in the magnitude relation.^[19]

Special Cases for Inscribed Polygons

Equilateral Triangle

In the context of Ptolemy's theorem applied to inscribed polygons, a notable special case arises when three vertices of a cyclic quadrilateral form an equilateral triangle. Consider an equilateral triangle ABC inscribed in a circle, with side length s. Let P be a point on the minor arc BC of the circumcircle (the arc not containing A). The quadrilateral ABPC is cyclic by construction, and Ptolemy's theorem applies to it, yielding the relation PA \cdot BC = PB \cdot CA + PC \cdot AB.^[2] Since ABC is equilateral, AB = BC = CA = s. Substituting these equalities into the equation simplifies it to PA \cdot s = PB \cdot s + PC \cdot s, or equivalently, PA = PB + PC. This identity holds specifically because the equal side lengths allow the cancellation in the Ptolemy relation.^[2] Geometrically, this result reflects the symmetry of the equilateral triangle and the uniform distribution of points on the circumcircle. The circumradius R of the equilateral triangle is s / \sqrt{3}, ensuring all points lie on the same circle, which is essential for the cyclic property. The identity PA = PB + PC provides a direct verification of Ptolemy's theorem in this degenerate polygonal case, where the fourth "side" is effectively bridged by the position of P on the arc. For points P elsewhere on the circumcircle, the more general form states that the sum of the two shorter distances among PA, PB, and PC equals the longest one, maintaining the theorem's integrity across positions.^[2]

Square

Consider a square ABCD with side length s. Since a square is cyclic, Ptolemy's theorem applies, where the sides are AB = BC = CD = DA = s and the diagonals are AC = BD = s\sqrt{2}.^[1]^[5] Substituting into Ptolemy's theorem, which states that for a cyclic quadrilateral, the product of the diagonals equals the sum of the products of the opposite sides, yields:

(s\sqrt{2}) \cdot (s\sqrt{2}) = s \cdot s + s \cdot s

Simplifying both sides gives $2s^2 = 2s^2, verifying that the identity holds true. This demonstrates how the theorem accommodates the equal lengths of opposite sides in a square.^[1]^[20] The result directly relates to the Pythagorean theorem, as each diagonal serves as the hypotenuse of a 45-45-90 right triangle formed by two adjacent sides of the square, confirming (s\sqrt{2})^2 = s^2 + s^2.^[5]^[1]

Rectangle

In a rectangle ABCD with adjacent sides of lengths a and b, the diagonals AC and BD are equal in length, each measuring \sqrt{a^2 + b^2}.^[21] Every rectangle is a cyclic quadrilateral because the sum of each pair of opposite angles is $180^\circ, allowing it to be inscribed in a circle.^[22] As such, Ptolemy's theorem applies directly to this configuration. With AB = CD = a and AD = BC = b, the theorem yields the relation between the diagonals and sides:

AC \cdot BD = AB \cdot CD + AD \cdot BC

Substituting the known lengths gives

\sqrt{a^2 + b^2} \cdot \sqrt{a^2 + b^2} = a \cdot a + b \cdot b,

which simplifies to the tautological equation a^2 + b^2 = a^2 + b^2. This identity holds universally for rectangles, verifying the consistency of Ptolemy's theorem in this setting and demonstrating its reduction to a basic geometric equality.^[1]^[23] The circumcircle of a rectangle has its center at the intersection of the diagonals, which bisect each other.^[24] This central point equidistant from all four vertices underscores the rectangle's inherent symmetry and cyclicity. Furthermore, applying Ptolemy's theorem to a rectangle highlights its close connection to the Pythagorean theorem. Each diagonal divides the rectangle into two congruent right-angled triangles, where the legs are the sides a and b, and the hypotenuse is the diagonal \sqrt{a^2 + b^2}, thereby reinforcing the fundamental relation a^2 + b^2 = (\sqrt{a^2 + b^2})^2.^[5] The square serves as a special case of the rectangle when a = b.

Regular Pentagon

In a regular pentagon ABCDE inscribed in a circle, all sides are equal in length, denoted by s, and all diagonals are equal in length, denoted by d. Consider the cyclic quadrilateral ABCD formed by four consecutive vertices of the pentagon. The sides of this quadrilateral are AB = s, BC = s, CD = s, and DA = d (a diagonal of the pentagon), while its diagonals are AC = d and BD = d.^[25] Applying Ptolemy's theorem to quadrilateral ABCD yields the relation AC \cdot BD = AB \cdot CD + AD \cdot BC, which simplifies to d \cdot d = s \cdot s + d \cdot s, or d^2 = s^2 + s d. Dividing both sides by s^2 gives \left(\frac{d}{s}\right)^2 = 1 + \frac{d}{s}. Letting \phi = \frac{d}{s}, the equation becomes \phi^2 = \phi + 1. Solving this quadratic equation, \phi = \frac{1 \pm \sqrt{5}}{2}, and taking the positive root (since lengths are positive), yields \phi = \frac{1 + \sqrt{5}}{2}, the golden ratio.^[25] This result confirms that the ratio of a diagonal to a side in a regular pentagon is the golden ratio \phi, satisfying the fundamental identity \phi^2 = \phi + [1](/page/1). The application demonstrates how Ptolemy's theorem elegantly derives this classical proportion without relying on trigonometric or coordinate methods.^[25]

Regular Decagon

In a regular decagon inscribed in a unit circle, the side length s subtends a central angle of $36^\circ, while the diagonals span 2, 3, or 4 sides, subtending central angles of $72^\circ, $108^\circ, and $144^\circ, respectively; their lengths are d_2 = 2 \sin 36^\circ, d_3 = 2 \sin 54^\circ, and d_4 = 2 \sin 72^\circ, with s = 2 \sin 18^\circ.^[26] To relate these using Ptolemy's theorem, consider the cyclic quadrilateral formed by four vertices separated such that the sides are s, d_2, s, and d_4, with diagonals both equal to d_3; labeling the vertices sequentially as A_1, A_2, A_4, A_5, the sides are A_1A_2 = s, A_2A_4 = d_2, A_4A_5 = s, and A_5A_1 = d_4, while the diagonals are A_1A_4 = d_3 and A_2A_5 = d_3.^[26] Applying Ptolemy's theorem yields d_3 \cdot d_3 = s \cdot s + d_2 \cdot d_4, or d_3^2 = s^2 + d_2 d_4. Substituting the chord length expressions gives \left(2 \sin 54^\circ\right)^2 = \left(2 \sin 18^\circ\right)^2 + \left(2 \sin 36^\circ\right) \left(2 \sin 72^\circ\right), simplifying to \sin^2 54^\circ = \sin^2 18^\circ + \sin 36^\circ \sin 72^\circ; since \sin 54^\circ = \cos 36^\circ and \sin 72^\circ = 2 \sin 36^\circ \cos 36^\circ, this identity relates the side and diagonals while facilitating derivations involving \cos 18^\circ through further trigonometric manipulation.^[26] Ptolemy employed analogous applications of his theorem in the Almagest to construct his table of chords, where the chord of $36^\circ represents the side of a regular decagon inscribed in a circle of diameter 120 parts, enabling computations of astronomical positions.^[27]

General Corollaries

Pythagoras's Theorem

Ptolemy's theorem provides a generalization of the Pythagorean theorem for cyclic quadrilaterals, where the latter emerges as a special case when the quadrilateral is a rectangle. Consider a rectangle ABCD with opposite sides equal and all interior angles measuring $90^\circ. Label the sides such that AB = CD = a and AD = BC = b, while the diagonals AC = BD = c. A rectangle is inherently cyclic, as its vertices lie on a circle with the intersection of the diagonals as the center, since the diagonals are equal in length and bisect each other.^[28]^[29] Applying Ptolemy's theorem to this cyclic quadrilateral yields the relation AC \cdot BD = AB \cdot CD + AD \cdot BC. Substituting the equal lengths gives c \cdot c = a \cdot a + b \cdot b, which simplifies to

c^2 = a^2 + b^2.

This equation precisely states the Pythagorean theorem, where c is the hypotenuse (diagonal) of the right triangle formed by sides a and b. The derivation relies on the supplementary nature of opposite angles in a cyclic quadrilateral (here, $90^\circ + 90^\circ = 180^\circ) and the equality of opposite sides, demonstrating how Ptolemy's theorem directly encompasses the right-triangle case without requiring additional geometric constructions.^[28] This algebraic application highlights the generality of Ptolemy's theorem, as the Pythagorean relation holds specifically for right-angled configurations embedded in a cyclic quadrilateral like the rectangle, unifying the two under a broader framework for planar geometry. The uniqueness of this derivation lies in its avoidance of limiting processes or triangle-specific axioms, instead leveraging the full structure of Ptolemy's statement to recover the classic result.^[28]

Law of Cosines

One approach to deriving the law of cosines from Ptolemy's theorem involves constructing a cyclic quadrilateral ABCD where the angle ∠ABC = θ and the opposite angle ∠ADC = 180° − θ, leveraging the property that opposite angles in a cyclic quadrilateral are supplementary. In this configuration, let AB = c, BC = a, CD = x, DA = y, with diagonals AC = b and BD = d. Ptolemy's theorem then gives the relation b d = c x + a y. This setup allows the law of cosines to be established for triangle ABC, where θ is the angle at B and b is the side opposite θ. To proceed, apply the law of sines in triangles ABC and ADC. In triangle ABC, b / sin θ = 2R, where R is the circumradius of the circle. In triangle ADC, the angle at D is 180° − θ, so sin(180° − θ) = sin θ, yielding b / sin(180° − θ) = 2R, confirming the shared circumcircle. Expressing the other sides using the law of sines provides b = 2R sin θ in both triangles, enabling the diagonals and sides to be related through trigonometric expressions. The derivation combines Ptolemy's relation with these sine expressions and considers the areas of the triangles to isolate the cosine term. The area of triangle ABC is (1/2) a c sin θ, and the area of triangle ADC is (1/2) x y sin(180° − θ) = (1/2) x y sin θ. Substituting the side lengths from the law of sines into Ptolemy's equation and manipulating algebraically—specifically, squaring the diagonal d and incorporating the cosine via the supplementary angle property—yields d² = c² + a² − 2 a c cos θ. This establishes the law of cosines for the diagonal BD opposite angle θ in the effective triangle formed by the configuration, and the result extends generally to any triangle by analogous construction on its circumcircle.

Compound Angle Formulas

Ptolemy's theorem provides a geometric basis for deriving the compound angle formulas for sine and cosine through the construction of a cyclic quadrilateral inscribed in a circle with diameter equal to 1. In this setup, angles A and B are positioned at adjacent vertices, and the sides are assigned lengths corresponding to the trigonometric functions of these angles: specifically, two opposite sides are \sin A and \cos B, while the other two are \cos A and \sin B. These lengths arise from the chord length formula in a circle of radius $1/2, where a chord subtending a central angle of $2\theta measures \sin \theta.^[30]^[31] One diagonal of the quadrilateral coincides with the diameter, thus having length 1, while the other diagonal subtends a central angle of $2(A + B), making its length \sin(A + B). Applying Ptolemy's theorem, which states that for a cyclic quadrilateral, the product of the diagonals equals the sum of the products of the opposite sides, yields:

$1 \cdot \sin(A + B) = (\sin A)(\cos B) + (\cos A)(\sin B).

Thus, the addition formula is obtained:

\sin(A + B) = \sin A \cos B + \cos A \sin B.

This derivation relies on the law of sines to relate the side lengths to the angles in the inscribed triangles.^[30]^[31] For the subtraction formula, a similar cyclic quadrilateral is configured with the same diameter of length 1, but adjusted so that the relevant diagonal subtends a central angle of $2(A - B), leading to its length \sin(A - B). Ptolemy's theorem then gives:

\sin(A - B) = \sin A \cos B - \cos A \sin B.

The cosine addition formula follows by considering the complementary angle relation, \cos(A + B) = \sin\left(\frac{\pi}{2} - (A + B)\right), and substituting the sine addition formula:

\cos(A + B) = \cos A \cos B - \sin A \sin B.

These derivations demonstrate how Ptolemy's theorem encapsulates the trigonometric identities for sums and differences of angles in a purely geometric framework.^[30]^[32]

Extensions

Ptolemy's Inequality

Ptolemy's inequality provides a generalization of Ptolemy's theorem to arbitrary quadrilaterals in the Euclidean plane. For any four points A, B, C, and D forming a quadrilateral ABCD, the product of the lengths of the diagonals satisfies

AC \cdot BD \leq AB \cdot CD + AD \cdot BC,

with equality holding if and only if the points are concyclic (lie on a common circle) or collinear (degenerate case).^[4] This inequality measures the extent of "non-cyclicity" in the quadrilateral through the deficit (AB \cdot CD + AD \cdot BC) - AC \cdot BD \geq 0; a positive deficit indicates the points do not lie on a circle, while zero deficit corresponds to the cyclic configuration where the diagonal product achieves its maximum value relative to the sides.^[4] As an illustrative example, consider a non-cyclic parallelogram ABCD with opposite sides AB = CD = 3, AD = BC = 4, and \angle BAD = 60^\circ (not a right angle, so not cyclic). The diagonals are AC = \sqrt{37} (computed via the law of cosines in \triangle ABC with \angle ABC = 120^\circ) and BD = \sqrt{13} (in \triangle ABD with \angle BAD = 60^\circ), yielding AC \cdot BD = \sqrt{481} \approx 21.93. The right-hand side is AB \cdot CD + AD \cdot BC = 9 + 16 = 25 > 21.93, confirming the strict inequality. In this sense, Ptolemy's inequality extends the theorem by establishing that, for fixed side lengths, the product of the diagonals is maximized precisely when the quadrilateral is cyclic.^[4]

Converse of Ptolemy's Theorem

The converse of Ptolemy's theorem states that a quadrilateral ABCD is cyclic if the product of its diagonals equals the sum of the products of its opposite sides, that is, if AC \cdot BD = AB \cdot CD + AD \cdot BC.^[33] This condition characterizes cyclic quadrilaterals among all quadrilaterals with given side and diagonal lengths.^[4] A proof of the converse can be sketched using the trigonometric form of the law of cosines applied to the triangles formed by one diagonal, say AC. In triangles ABC and ADC, the law of cosines gives expressions involving the cosines of the angles at B and D. Substituting into the given equality yields \cos B + \cos D = 0, which implies that angles B and D are supplementary (B + D = \pi). Since opposite angles are supplementary, the quadrilateral is cyclic.^[33] This theorem provides a practical test for cyclicity when the lengths of all sides and diagonals are known, allowing one to verify without constructing the figure or measuring angles. It finds application in competition geometry, such as problems in the International Mathematical Olympiad (IMO) and American Invitational Mathematics Examination (AIME), where determining if a quadrilateral is cyclic aids in solving for unknown lengths or angles.^[34] The theorem assumes a convex quadrilateral; for non-convex quadrilaterals, the equality may hold without the points being concyclic, as the configuration can degenerate or cross.^[4]

Theorem on the Ratio of Diagonals

In a cyclic quadrilateral ABCD, the theorem on the ratio of diagonals states that the ratio of the lengths of the diagonals AC to BD is equal to the ratio of the sums of the products of the pairs of sides sharing the endpoints of each diagonal:

\frac{AC}{BD} = \frac{AB \cdot AD + BC \cdot CD}{AB \cdot BC + AD \cdot CD}.

This relation holds specifically for cyclic quadrilaterals and provides a way to compare the diagonals using only the side lengths, building on the product equality from Ptolemy's theorem.^[35]^[36] A derivation of this ratio can be sketched using properties of the intersecting diagonals. Let P be the intersection point of diagonals AC and BD in cyclic quadrilateral ABCD. Triangles APD and BP C are similar because they subtend equal angles due to the cyclic nature (∠APD = ∠BPC as vertical angles, and ∠PAD = ∠PBC, ∠PDA = ∠PCB as inscribed angles). From similarity, ratios such as AP/BP = AD/BC = DP/CP hold, leading to equalities like AB · AD / AP = BC · CD / CP. Combining these proportions and solving for the full diagonal segments yields AC/BD as the specified ratio after cancellation.^[35]^[36] Geometrically, the theorem interprets the diagonal ratio as a balance between the products of adjacent sides at opposite vertices: the numerator aggregates products from vertices A and C (endpoints of AC), while the denominator aggregates those from B and D (endpoints of BD). This reflects how the cyclic condition distributes side influences symmetrically across the diagonals.^[35] For verification, consider a rectangle ABCD (a special cyclic quadrilateral) with AB = CD = l (length) and AD = BC = w (width). The diagonals AC and BD are equal, so AC/BD = 1. Substituting into the formula gives (AB · AD + BC · CD) / (AB · BC + AD · CD) = (l · w + w · l) / (l · w + w · l) = (2lw) / (2lw) = 1, confirming the relation.^[35]

Applications in Trigonometry

Ptolemy utilized his theorem to develop chord tables in the Almagest, representing sine values for angles in a unit circle through recursive computations on inscribed quadrilaterals. By applying the theorem to cyclic quadrilaterals formed by arcs on the circle, he derived chord lengths for sums and differences of known arcs, enabling systematic calculation of chords at half-degree intervals from ½° to 180°. This approach, starting from basic angles like 36° and 72°, facilitated trigonometric evaluations essential for astronomical modeling without direct angle measurements.^[27] The recursive application of Ptolemy's theorem in these quadrilaterals directly yields trigonometric addition formulas, as detailed in Almagest Book I. For instance, the chord of the sum of two arcs corresponds to the formula \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta, while the difference formula emerges similarly, providing a geometric basis for compound angle identities used in celestial calculations. These identities supported derivations for solar, lunar, and planetary positions by relating arc lengths to observable angles.^[11]^[27] In contemporary settings, Ptolemy's theorem aids computational geometry by verifying if four points are concyclic: the points lie on a circle if and only if the product of one pair of opposite sides plus the product of the other pair equals the product of the diagonals in the formed quadrilateral. This check is efficient for algorithms involving circle fitting or geometric constraints. Additionally, the theorem features in mathematical Olympiad problems, where it resolves trigonometric challenges in cyclic configurations, such as evaluating side lengths or angles without coordinate geometry.

References

[1]
Ptolemy's Theorem -- from Wolfram MathWorld
For a cyclic quadrilateral, the sum of the products of the two pairs of opposite sides equals the product of the diagonals AB×CD+BC×DA=AC×BD (1) (Kimberling ...
[2]
Ptolemy's Theorem - Interactive Mathematics Miscellany and Puzzles
Let a convex quadrilateral ABCD be inscribed in a circle. Then the sum of the products of the two pairs of opposite sides equals the product of its two ...<|control11|><|separator|>
[3]
[PDF] History of Mathematics: Ptolemy's Theorem - Parabola
Both Hipparchus and Menelaus produced chord-tables, so it is possible that their (poorly preserved) texts included accounts of Ptolemy's Theorem. However, ...
[4]
[PDF] Ptolemy's Theorem and its converse - UChicago Math
We show that 4 points in the plane lie on a circle or straight line if and only if they satisfy Ptolemy's condition. 1. The Theorems. If A and B are points in ...
[5]
Ptolemy's Theorem | Brilliant Math & Science Wiki
Ptolemy's theorem states the relationship between the diagonals and the sides of a cyclic quadrilateral. It is a powerful tool to apply to problems about ...
[6]
What Is Ptolemy's Theorem? - ScienceABC
Apr 4, 2020 · ... Ptolemy's Theorem, imagine a rectangle ABCD inscribed inside a circle. Applying Ptolemy's Theorem to rectangle ABCD, we have. AD⋅BC = AB⋅DC ...
[7]
[PDF] 11 · The Culmination of Greek Cartography in Ptolemy
While it is generally believed that Ptolemy was born in Upper Egypt and subsequently lived in Alexandria, he is known to us mainly through his various writings ...
[8]
Claudius Ptolemaeus - Ptolemy Project
Ptolemy was an astronomer, mathemetician and geographer in the second century AD. He codified the Greek geocentric view of the universe.Missing: biography | Show results with:biography
[9]
Mathematical tables in Ptolemy's Almagest - ScienceDirect
Feb 26, 2014 · This paper is a discussion of Ptolemy's use of mathematical tables in the Almagest. By focusing on Ptolemy's mathematical practice and ...
[10]
[PDF] Ptolemy Finds High Noon in Chords of Circles
Dec 22, 2020 · What's more, Ptolemy made many other uses of his table of chords in the Almagest to compute solutions to other astronomical problems. The ...
[11]
[PDF] Ptolemy's ALMAGEST - Classical Liberal Arts Academy
In the course of making the translation I recomputed all the numerical results in the text, and all the tables (the latter mostly by means of computer programs) ...
[12]
https://ptolemaeus.badw.de/work/19
[13]
ṬUSI, NAṢIR-AL-DIN ii. AS MATHEMATICIAN AND ASTRONOMER
Jul 20, 2009 · In his redaction of the Almagest we saw that he replaced the two chord theorems of the Almagest, those of Menelaus and Ptolemy, with their ...
[14]
Gerard of Cremona's Latin translation of the Almagest and the ...
Feb 4, 2023 · The first complete, printed edition of Ptolemy's Almagest appeared in 1515 and is based on Gerard of Cremona's translation from Arabic into ...
[15]
Derivation / Proof of Ptolemy's Theorem for Cyclic Quadrilateral
... cos(180∘–β). d12=c2+d2+2cdcosβ → equation (2). Equate ... cosβ+2cdcosβ. a2+b2–c2−d2=2(ab+cd) ...
[16]
[PDF] Inversions. Popular L'Oeures in Mathematics. 7Sp. - ERIC
The method of inversions allows us to find these solutions easily. Let 7 he an inversion with center V 'and radius r such that thc circle of inversion ...
[17]
Ptolemy via Cross-Ratio
Sep 9, 2011 · Ptolemy's theorem is equivalent to the fact that for cross ratios ... For example, the cross-ratio (abcd) is the quotient of two single ...
[18]
None
### Summary: Applying Ptolemy's Theorem to a Square
[19]
Rectangle -- from Wolfram MathWorld
A closed planar quadrilateral with opposite sides of equal lengths a and b, and with four right angles. A square is a degenerate rectangle with a=b.
[20]
Definitions and Hierarchy of Quadrilaterals - UTSA
Dec 12, 2021 · Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A convex quadrilateral is cyclic if and only if opposite angles sum to ...
[21]
Ptolemy's theorem - MacTutor - University of St Andrews
Ptolemy's theorem. For a cyclic quadrilateral, the sum of the products of the ... If A B C D ABCD ABCD is a rectangle, this reduces to Pythagoras's theorem.
[22]
[PDF] Inscribed (Cyclic) Quadrilaterals and Parallelograms
Students will be able to identify a rectangle as the only parallelogram that can be inscribed in a circle.
[23]
[PDF] How to Prove It
We apply Ptolemy's theorem to the quadrilateral ABDC: AB · CD + AC · BD = AD · BC. Since AB = AC = √2, BD = 2cos x, CD = 2sin x, BC = 2, we get: 2 √ 2 sin x + ...
[24]
Ptolemy's theorem - Scientific Library
Ptolemy's theorem. In Euclidean geometry, Ptolemy's theorem is a relation ... regular decagon (length c) and the fourth DG is of length z. Diagonals DC ...
[25]
Ptolemy's Table of Chords: Trigonometry in the Second Century
A historical paper on the creation of a Second Century trig table, outlining the procedure and detailing it using modern notation.
[26]
Pythagoras's Theorem/Proof 8 - ProofWiki
### Summary of Pythagoras's Theorem/Proof 8
[27]
None
### Summary: Application of Ptolemy's Theorem to Prove Pythagoras Theorem
[28]
Sine, Cosine, and Ptolemy's Theorem
The Law of Sines supplies the length of the remaining diagonal. The addition formula for sine is just a reformulation of Ptolemy's theorem. To prove the ...
[29]
Derive Sum of Two Angles (Ptolemy's Theorem) - Wumbo
This example derives the sum of two angle identities using the law of sines, the inscribed angle theorem and Ptolemy's Theorem.
[30]
Deriving the Composite Angle Formulae for Sine from Ptolemy
By applying. Ptolemy's theorem, we obtain: sin A cos B + cos A sin B = 1.sin (A + B) = sin (A + B). Similarly, the difference formula can be obtained from the ...Missing: addition | Show results with:addition
[31]
[PDF] MORE CHARACTERIZATIONS OF CYCLIC QUADRILATERALS
Theorem 2.1. In a convex quadrilateral ABCD, the equality sinα sinγ = sinβ sinδ holds if and only if it is a cyclic quadrilateral.
[32]
[PDF] 2024-11-09: Chapel Hill Math Circle on Ptolemy's Theorem
Nov 9, 2024 · Ptolemy's Theorem is a result relating the lengths of the sides and diagonals in a cyclic · quadrilateral, meaning a quadrilateral all of whose ...
[33]
Diagonals in Cyclic Quadrilateral
In a cyclic quadrilateral ABCD the ratio of the diagonals equals the ratio of the sums of products of the sides that share the diagonals' end points.
[34]
None
### Summary of Proof of Ptolemy’s Theorem Using Similar Triangles
[35]
[PDF] A Treatise On Spherical Trigonometry, And Its Application To ...
This follows from Ptolemy's theorem, since chord BC = 2sin£2?C, &c. 6. In ... APPLICATIONS OF SPHERICAL TRIGONOMETRY TO GEODESY. AND ASTRONOMY. Section ...