Ptolemy's theorem

In Euclidean geometry, Ptolemy's theorem is a special rule about the sides and diagonals of a cyclic quadrilateral. A cyclic quadrilateral is a four-sided shape where all the corners, or vertices, fit on a circle. This idea is named after the Greek astronomer and mathematician Ptolemy.

Ptolemy used this theorem to help study the stars and planets. The theorem tells us that for any cyclic quadrilateral with corners labeled A, B, C, and D, a special math rule connects the lengths of the sides and the diagonals.

The rule says that multiplying the lengths of the two diagonals together equals the sum of multiplying the lengths of each pair of opposite sides. If this rule is true for any four-sided shape, then that shape can be drawn on a circle, making it a cyclic quadrilateral. This theorem helps us solve many geometry problems and understand shapes better.

Corollaries on inscribed polygons

Equilateral triangle

Ptolemy's Theorem helps us understand shapes like equilateral triangles inside circles. If you have an equilateral triangle inside a circle and choose a point on the circle, the distance from that point to the farthest corner of the triangle is the same as the total of its distances to the two closer corners.

Square

A square can also be drawn inside a circle. The center of the circle is the same as the center of the square. If each side of the square measures a, the diagonal across the square measures a√2. This matches what we find using Ptolemy's ideas.

Rectangle

For any rectangle inside a circle, the same idea works. If the rectangle has sides a and b and diagonal d, then d² equals a² + b². This is the same as the Pythagorean theorem.

Pentagon

For a regular pentagon inside a circle, there is a special relationship between the length of its sides (a) and the length of its inner lines (b). This relationship involves the golden ratio, a number that appears often in nature and art.

Side of decagon

If we draw a decagon (a 10-sided shape) inside a circle, we can use Ptolemy's ideas again. By looking at certain lines and distances, we can find the length of each side of the decagon using the diameter of the circle and the golden ratio.

Proofs

Further information: Proofs of trigonometric identities

Visual proof

The animation here shows how Ptolemy's theorem can be shown with a picture.

Proof by similarity of triangles

Let ABCD be a cyclic quadrilateral. We make a point K on AC so that ∠ABK = ∠CBD. By looking at the angles, △ABK is similar to △DBC, and △ABD is similar to △KBC. This means AK/AB = CD/BD, and CK/BC = DA/BD. We can write this as AK⋅BD = AB⋅CD, and CK⋅BD = BC⋅DA. Adding these together, we get AK⋅BD + CK⋅BD = AB⋅CD + BC⋅DA. Factoring this gives (AK+CK)·BD = AB⋅CD + BC⋅DA. Since AK+CK = AC, we have AC⋅BD = AB⋅CD + BC⋅DA.

This proof works for simple cyclic quadrilaterals. If the quadrilateral crosses itself, K will be outside AC, but the result still works.

Proof by the Simson line

We use a rule that says for a point D and triangle △ABC, the feet of the lines from D to the sides of the triangle are in a line. We place D on the circle around △ABC to make the cyclic quadrilateral ABCD. Then the feet are in a line, called the Simson line.

Proof by trigonometric identities

Let the angles at the points where AB, BC, and CD meet the circle be α, β, and γ. Let the circle’s radius be R. Then AB = 2R sin α, BC = 2R sin β, CD = 2R sin γ, and so on. We need to show that sin (α + β) sin (β + γ) equals sin α sin γ + sin β sin (α + β + γ). Using rules for adding angles, both sides become the same thing.

Here’s a simpler way. Imagine a new quadrilateral ABCD' with the same circle, where A, B, C stay the same, but D' is moved so that |AD'| = |CD| and |CD'| = |AD|. Then ABCD' has the same side lengths and angles as ABCD, just in a different order.

Proof by inversion

We pick a helper circle Γ centered at D. We change the view so the big circle becomes a straight line. Then we show that A'B' + B'C' = A'C'. We can write A'B', B'C', and A'C' in terms of the original points and show they match Ptolemy’s rule.

Proof using complex numbers

We put ABCD on the complex plane by matching A to z_A, B to z_B, and so on. We make a special number called the cross-ratio.

We then show that AB⋅CD + AD⋅BC equals AC⋅BD, which proves Ptolemy's inequality when the points are on a circle.

Corollaries

When we have a circle with a diameter of one unit, the lengths of the sides of any four-sided shape on the circle match the sine values of some angles. The lengths of the diagonals match the sine of the sum of pairs of these angles. Using this idea, we can rewrite Ptolemy’s Theorem in a form that uses trigonometry.

By applying specific conditions to these angles, we can create useful results from this starting point. Remember that the total of all four angles in this shape always adds up to 180 degrees.

Corollary 1. Pythagoras's theorem

If two pairs of angles are equal, we can show a relationship that matches Pythagoras’s famous rule about squares and triangles.

Corollary 2. The law of cosines

When two angles are the same, the shape becomes a special kind of quadrilateral. This helps us understand how to measure sides and angles in triangles.

Corollary 3. Compound angle sine (+)

If the total of two pairs of angles each equals 90 degrees, we can find a way to combine sines of angles.

Corollary 4. Compound angle sine (−)

If one angle is exactly 90 degrees, we can use this to find relationships between sines and cosines of combined angles.

Corollary 5. Compound angle cosine (+)

If one angle is 90 degrees, we can also find a way to combine cosines of angles.

Even without the easy math symbols we use today, Ptolemy’s work gave ancient scholars strong tools to create accurate charts of angles. These charts helped them study the stars and the universe. Earlier astronomers likely used similar ideas, meaning the origins of these methods go back even further into history.

Ptolemy's inequality

Main article: Ptolemy's inequality

Ptolemy's inequality helps us learn about shapes that are not perfect circles. It shows us that for any shape with four points, a special math rule will always work. This rule is perfect and exactly right only when the four points can be placed on a circle, which is when Ptolemy's theorem is used.

Related theorem about the ratio of the diagonals

Ptolemy’s theorem helps us find the size of the diagonals in a special shape called a cyclic quadrilateral by looking at the sides. There is another theorem that helps us find the comparison of the lengths of these diagonals.

To understand this, we look at the area of triangles formed inside the circle. By adding the areas of two triangles that make up the quadrilateral in two different ways, we can find a special formula. This formula tells us how the lengths of the diagonals compare to each other.

With both the product and the ratio of the diagonals, we can then find the exact lengths of each diagonal.