Ptolemy's theorem

Key takeaways

Ptolemy's theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Ptolemy's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Ptolemy's theorem from memory before moving on to harder problems.

Reference excerpt

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). 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:

A C ⋅ B D = A B ⋅ C D + B C ⋅ A D {\displaystyle AC\cdot BD=AB\cdot CD+BC\cdot AD}

This relation may be verbally expressed as follows:

If a quadrilateral is cyclic then the product of the lengths of its diagonals is equal to the sum of the products of the lengths of the pairs of opposite sides. Moreover, the converse of Ptolemy's theorem is also true:

In a quadrilateral, if the sum of the products of the lengths of its two pairs of opposite sides is equal to the product of the lengths of its diagonals, then the quadrilateral can be inscribed in a circle i.e. it is a cyclic quadrilateral. 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:

q s = p s + r s ⇒ q = p + r . {\displaystyle qs=ps+rs\Rightarrow q=p+r.}

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 a {\displaystyle a} then the length of the diagonal is equal to a 2 {\displaystyle a{\sqrt {2}}} 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:

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. By completing the square, the relation yields the golden ratio:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Ptolemy's theorem

Start with the simplest possible case. Write down what Ptolemy's theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Ptolemy's theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Ptolemy's theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Ptolemy's theorem

In research: Ptolemy's theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Ptolemy's theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Ptolemy's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Ancient Greek mathematics, Euclidean plane geometry, Ptolemy, so understanding it makes those chapters shorter.
In everyday life: Look for Ptolemy's theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Ptolemy's theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ptolemy's theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Ptolemy's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ptolemy's theorem in simple terms?

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus).

Why does Ptolemy's theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Ptolemy's theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Ptolemy's theorem.