Ptolemy's theorem

Equilateral triangle

Ptolemy's Theorem yields as a corollary a pretty 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:

q s = p s + r s ⇒ q = p + r .

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 a then the length of the diagonal is equal to a 2 according to the Pythagorean theorem and the 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:

Furthermore it is clear (manifestum est) that when the chord subtending an arc has been given, that chord too can be found which subtends the rest of the semicircle.

Pentagon

A more interesting example is the relation between the length a of the side and the (common) length b of the 5 chords in a regular pentagon. In this case the relation reads b² = a² + ab which yields the golden ratio

φ = b a = 1 + 5 2 .

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:

a d = 2 b c ⇒ a d = 2 φ a c where φ is the golden ratio. ⇒ c = d 2 φ .

whence the side of the inscribed decagon is obtained in terms of the circle diameter. Pythagoras' 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 (following Ptolemy) wrote,

"The diameter of a circle being given, the sides of the triangle, tetragon, pentagon, hexagon and decagon, which the same circle circumscribes, are also given."

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 △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 (AK+CK)·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 trigonometric identities

Let the inscribed angles subtended by A B , B C and C D be, respectively, α , β and γ , and the radius of the circle be R , then we have A B = 2 R sin ⁡ α , B C = 2 R sin ⁡ β , C D = 2 R sin ⁡ γ , A D = 2 R sin ⁡ ( α + β + γ ) , A C = 2 R sin ⁡ ( α + β ) and B D = 2 R sin ⁡ ( β + γ ) , and the original equality to be proved is transformed to

sin ⁡ ( α + β ) sin ⁡ ( β + γ ) = sin ⁡ α sin ⁡ γ + sin ⁡ β sin ⁡ ( α + β + γ )

from which the factor 4 R 2 has disappeared by dividing both sides of the equation by it.

Now by using the sum formulae, sin ⁡ ( x + y ) = sin ⁡ x cos ⁡ y + cos ⁡ x sin ⁡ y and cos ⁡ ( x + y ) = cos ⁡ x cos ⁡ y − sin ⁡ x sin ⁡ y , it is trivial to show that both sides of the above equation are equal to

sin ⁡ α sin ⁡ β cos ⁡ β cos ⁡ γ + sin ⁡ α cos 2 ⁡ β sin ⁡ γ + cos ⁡ α sin 2 ⁡ β cos ⁡ γ + cos ⁡ α sin ⁡ β cos ⁡ β sin ⁡ γ .

Q.E.D.

Corollaries

In the case of a circle of unit diameter the sides S 1 , S 2 , S 3 , S 4 of any cyclic quadrilateral ABCD are numerically equal to the sines of the angles θ 1 , θ 2 , θ 3 and θ 4 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:

sin ⁡ θ 1 sin ⁡ θ 3 + sin ⁡ θ 2 sin ⁡ θ 4 = sin ⁡ ( θ 3 + θ 2 ) sin ⁡ ( θ 1 + θ 2 )

Applying certain conditions to the subtended angles θ 1 , θ 2 , θ 3 and θ 4 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 θ 1 + θ 2 + θ 3 + θ 4 = 180 ∘ .

Corollary 1. Pythagoras' theorem

Let θ 1 = θ 3 and θ 2 = θ 4 . Then θ 1 + θ 2 = θ 3 + θ 4 = 90 ∘ (since opposite angles of a cyclic quadrilateral are supplementary). Then:

sin ⁡ θ 1 sin ⁡ θ 3 + sin ⁡ θ 2 sin ⁡ θ 4 = sin ⁡ ( θ 3 + θ 2 ) sin ⁡ ( θ 3 + θ 4 ) sin 2 ⁡ θ 1 + sin 2 ⁡ θ 2 = sin 2 ⁡ ( θ 1 + θ 2 ) sin 2 ⁡ θ 1 + cos 2 ⁡ θ 1 = 1

Corollary 2. The law of cosines

Let θ 2 = θ 4 . 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 2 x units where:

x = S 2 cos ⁡ ( θ 2 + θ 3 )

It will be easier in this case to revert to the standard statement of Ptolemy's theorem:

S 1 S 3 + S 2 S 4 = A C ¯ ⋅ B D ¯ ⇒ S 1 S 3 + S 2 2 = A C ¯ 2 ⇒ S 1 [ S 1 − 2 S 2 cos ⁡ ( θ 2 + θ 3 ) ] + S 2 2 = A C ¯ 2 ⇒ S 1 2 + S 2 2 − 2 S 1 S 2 cos ⁡ ( θ 2 + θ 3 ) = A C ¯ 2

The cosine rule for triangle ABC.

Corollary 3: Compound angle sine (+)

Let

θ 1 + θ 2 = θ 3 + θ 4 = 90 ∘ .

Then

sin ⁡ θ 1 sin ⁡ θ 3 + sin ⁡ θ 2 sin ⁡ θ 4 = sin ⁡ ( θ 3 + θ 2 ) sin ⁡ ( θ 3 + θ 4 )

Therefore,

cos ⁡ θ 2 sin ⁡ θ 3 + sin ⁡ θ 2 cos ⁡ θ 3 = sin ⁡ ( θ 3 + θ 2 ) × 1

Formula for compound angle sine (+).

Corollary 4: Compound angle sine (−)

Let θ 1 = 90 ∘ . Then θ 2 + ( θ 3 + θ 4 ) = 90 ∘ . Hence,

sin ⁡ θ 1 sin ⁡ θ 3 + sin ⁡ θ 2 sin ⁡ θ 4 = sin ⁡ ( θ 3 + θ 2 ) sin ⁡ ( θ 3 + θ 4 ) sin ⁡ θ 3 + sin ⁡ θ 2 cos ⁡ ( θ 2 + θ 3 ) = sin ⁡ ( θ 3 + θ 2 ) cos ⁡ θ 2 sin ⁡ θ 3 = sin ⁡ ( θ 3 + θ 2 ) cos ⁡ θ 2 − cos ⁡ ( θ 2 + θ 3 ) sin ⁡ θ 2

Formula for compound angle sine (−).

This derivation corresponds to the Third Theorem as chronicled by Copernicus following Ptolemy in Almagest. In particular if the sides of a pentagon (subtending 36° at the circumference) and of a hexagon (subtending 30° at the circumference) 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 Fifth Theorem as chronicled by Copernicus following Ptolemy in Almagest.

Let θ 3 = 90 ∘ . Then θ 1 + ( θ 2 + θ 4 ) = 90 ∘ . Hence

sin ⁡ θ 1 sin ⁡ θ 3 + sin ⁡ θ 2 sin ⁡ θ 4 = sin ⁡ ( θ 3 + θ 2 ) sin ⁡ ( θ 3 + θ 4 ) cos ⁡ ( θ 2 + θ 4 ) + sin ⁡ θ 2 sin ⁡ θ 4 = cos ⁡ θ 2 cos ⁡ θ 4 cos ⁡ ( θ 2 + θ 4 ) = cos ⁡ θ 2 cos ⁡ θ 4 − sin ⁡ θ 2 sin ⁡ θ 4

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 (or more simply the Second 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 (corresponding to tables of sines) 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

A B ¯ ⋅ C D ¯ + B C ¯ ⋅ D A ¯ ≥ A C ¯ ⋅ B D ¯

where equality holds if and only if the quadrilateral is cyclic. This special case is equivalent to Ptolemy's theorem.