Ptolemy's theorem


In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.
If the vertices of the cyclic quadrilateral are A, B, C, and D in order, then the theorem states that:
This relation may be verbally expressed as follows:
Moreover, the converse of Ptolemy's theorem is also true:
To appreciate the utility and general significance of Ptolemy’s Theorem, it is especially useful to study its main [|corollaries].

Corollaries on inscribed polygons

Equilateral triangle

Ptolemy's Theorem yields as a corollary a theorem regarding an equilateral triangle inscribed in a circle.
Given An equilateral triangle inscribed on a circle, and a point on the circle.
The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices.
Proof: Follows immediately from Ptolemy's theorem:
This corollary has as an application an algorithm for computing minimal Steiner trees whose topology is fixed, by repeatedly replacing pairs of leaves of the tree A, B that should be connected to a Steiner point, by the third point C of their equilateral triangle. The unknown Steiner point must lie on arc AB of the circle, and this replacement ensures that, no matter where it is placed, the length of the tree remains unchanged.

Square

Any square can be inscribed in a circle whose center is the center of the square. If the common length of its four sides is equal to then the length of the diagonal is equal to according to the Pythagorean theorem, and Ptolemy's relation obviously holds.

Rectangle

More generally, if the quadrilateral is a rectangle with sides a and b and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d2, the right hand side of Ptolemy's relation is the sum a2 + b2.
Copernicus – who used Ptolemy's theorem extensively in his trigonometrical work – refers to this result as a 'Porism' or self-evident corollary:

Pentagon

A more interesting example is the relation between the length a of the side and the length b of the 5 chords in a regular pentagon. By completing the square, the relation yields the golden ratio:

Side of decagon

If now diameter AF is drawn bisecting DC so that DF and CF are sides c of an inscribed decagon, Ptolemy's Theorem can again be applied – this time to cyclic quadrilateral ADFC with diameter d as one of its diagonals:
whence the side of the inscribed decagon is obtained in terms of the circle diameter. Pythagoras's theorem applied to right triangle AFD then yields "b" in terms of the diameter and "a" the side of the pentagon is thereafter calculated as
As Copernicus wrote,

Proofs

Visual proof

The animation here shows a visual demonstration of Ptolemy's theorem, based on Derrick & Herstein.

Proof by similarity of triangles

Let ABCD be a cyclic quadrilateral.
On the chord BC, the inscribed angles ∠BAC = ∠BDC, and on AB, ∠ADB = ∠ACB.
Construct K on AC such that ∠ABK = ∠CBD; since ∠ABK + ∠CBK = ∠ABC = ∠CBD + ∠ABD, ∠CBK = ∠ABD.
Now, by common angles, △ABK is similar to △DBC, and likewise △ABD is similar to △KBC.
Thus AK/AB = CD/BD, and CK/BC = DA/BD;
equivalently, AK⋅BD = AB⋅CD, and CK⋅BD = BC⋅DA.
By adding two equalities, we have AK⋅BD + CK⋅BD = AB⋅CD + BC⋅DA, and factorizing this gives ·BD = AB⋅CD + BC⋅DA.
But AK+CK = AC, so AC⋅BD = AB⋅CD + BC⋅DA, Q.E.D.
The proof as written is only valid for simple cyclic quadrilaterals. If the quadrilateral is self-crossing, then K will be located outside the line segment AC. But in this case, AK−CK = ±AC, giving the expected result.

Proof by the Simson line

We first use the Theorem stating that for distances from a fixed point to the vertices of, and perpendiculars dropped to the three sides, where the feet of these perpendiculars are the vertices of a triangle, the sides of are
where is the radius of the triangle circumcircle. To prove this, we apply the Law of Sines to the triangles,,, and also to itself.
In the sequel, we place the point on the circumcircle to obtain the cyclic quadrilateral. Now, the feet are colinear, and form the Simsone line. Thus,, and using yields Ptolemy's theorem.

Proof by trigonometric identities

Let the inscribed angles subtended by, and be, respectively,, and, and the radius of the circle be, then we have,,,, and, and the original equality to be proved is transformed to
from which the factor has disappeared by dividing both sides of the equation by it.
Now by using the angle sum formulae, and, it is trivial to show that both sides of the above equation are equal to
Q.E.D.
Here is another, perhaps more transparent, proof using rudimentary trigonometry.
Define a new quadrilateral inscribed in the same circle, where are the same
as in, and located at a new point on the same circle, defined by,
.
Then, has the same lengths of edges, and consequently the same inscribed angles subtended by
the corresponding edges, as, only in a different order. That is,, and, for, respectively, and.
Also, and have the same area. Then,
Q.E.D.

Proof by inversion

Choose an auxiliary circle of radius centered at D with respect to which the circumcircle of ABCD is inverted into a line.
Then
Then and can be expressed as
, and respectively. Multiplying each term by and using yields Ptolemy's equality.
Q.E.D.
Note that if the quadrilateral is not cyclic then A', B' and C' form a triangle and hence A'B'+B'C' > A'C', giving us a very simple proof of Ptolemy's Inequality which is presented below.

Proof using complex numbers

Embed ABCD in the complex plane by identifying as four distinct complex numbers. Define the cross-ratio
Then
with equality if and only if the cross-ratio is a positive real number. This proves Ptolemy's inequality generally, as it remains only to show that lie consecutively arranged
on a circle in if and only if.
From the polar form of a complex number, it follows
with the last equality holding if and only if ABCD is cyclic, since a quadrilateral is cyclic if and only if opposite angles sum to.
Q.E.D.
Note that this proof is equivalently made by observing that the cyclicity of ABCD, i.e. the supplementarity and, is equivalent to the condition
in particular there is a rotation of in which this is 0, and by which Ptolemy's theorem
is then directly established from the simple algebraic identity

Corollaries

In the case of a circle of unit diameter the sides of any cyclic quadrilateral ABCD are numerically equal to the sines of the angles and which they subtend. Similarly the diagonals are equal to the sine of the sum of whichever pair of angles they subtend. We may then write Ptolemy's Theorem in the following trigonometric form:
Applying certain conditions to the subtended angles and it is possible to derive a number of important corollaries using the above as our starting point. In what follows it is important to bear in mind that the sum of angles.

Corollary 1. Pythagoras's theorem

Let and. Then
. Then:

Corollary 2. The law of cosines

Let. The rectangle of corollary 1 is now a symmetrical trapezium with equal diagonals and a pair of equal sides. The parallel sides differ in length by units where:
It will be easier in this case to revert to the standard statement of Ptolemy's theorem:
The cosine rule for triangle ABC.

Corollary 3. Compound angle sine (+)

Let
Then
Therefore,
Formula for compound angle sine.

Corollary 4. Compound angle sine (−)

Let. Then. Hence,
Formula for compound angle sine .
This derivation corresponds to the
as chronicled by Copernicus following Ptolemy in Almagest. In particular if the sides of a pentagon and of a hexagon are given, a chord subtending 6° may be calculated. This was a critical step in the ancient method of calculating tables of chords.

Corollary 5. Compound angle cosine (+)

This corollary is the core of the as chronicled by Copernicus following Ptolemy in Almagest.
Let. Then. Hence
Formula for compound angle cosine
Despite lacking the dexterity of our modern trigonometric notation, it should be clear from the above corollaries that in Ptolemy's theorem the ancient world had at its disposal an extremely flexible and powerful trigonometric tool which enabled the cognoscenti of those times to draw up accurate tables of chords and to use these in their attempts to understand and map the cosmos as they saw it. Since tables of chords were drawn up by Hipparchus three centuries before Ptolemy, we must assume he knew of the 'Second Theorem' and its derivatives. Following the trail of ancient astronomers, history records the star catalogue of Timocharis of Alexandria. If, as seems likely, the compilation of such catalogues required an understanding of the 'Second Theorem', then the true origins of the latter disappear thereafter into the mists of antiquity; but it cannot be unreasonable to presume that the astronomers, architects and construction engineers of ancient Egypt may have had some knowledge of it.

Ptolemy's inequality

The equation in Ptolemy's theorem is never true with non-cyclic quadrilaterals. Ptolemy's inequality is an extension of this fact, and it is a more general form of Ptolemy's theorem. It states that, given a quadrilateral ABCD, then
where equality holds if and only if the quadrilateral is cyclic. This special case is equivalent to Ptolemy's theorem.

Related theorem about the ratio of the diagonals

Ptolemy's theorem gives the product of the diagonals knowing the sides. The following theorem yields the same for the ratio of the diagonals.
Proof: It is known that the area of a triangle inscribed in a circle of radius is:
Writing the area of the quadrilateral as sum of two triangles sharing the same circumscribing circle, we obtain two relations for each decomposition.
Equating, we obtain the announced formula.
Consequence: Knowing both the product and the ratio of the diagonals, we deduce their immediate expressions: