Ptolemy's theorem

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

Ptolemy used this theorem to help make his table of chords, which was a very important tool for studying 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 if you multiply the lengths of the two diagonals together, it will equal the sum of multiplying the lengths of each pair of opposite sides. Even more interesting, if this rule holds 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 problems in geometry and understand shapes better.

Corollaries on inscribed polygons

Equilateral triangle

Ptolemy's Theorem helps us understand shapes like equilateral triangles drawn inside circles. If you have an equilateral triangle inside a circle and pick a point on the circle, the distance from that point to the farthest corner of the triangle equals the sum 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 a visual demonstration of Ptolemy's theorem.

Proof by similarity of triangles

Let ABCD be a cyclic quadrilateral. Construct K on AC such that ∠ABK = ∠CBD. 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 (AK+CK)·BD = AB⋅CD + BC⋅DA. But AK+CK = AC, so AC⋅BD = AB⋅CD + BC⋅DA.

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 D to the vertices of △ABC, and perpendiculars dropped to the three sides, the feet of these perpendiculars are the vertices of a triangle. In the sequel, we place the point D on the circumcircle to obtain the cyclic quadrilateral ABCD. Now, the feet are colinear, and form the Simsone line.

Proof by trigonometric identities

Let the inscribed angles subtended by AB, BC and CD be, respectively, α, β and γ, and the radius of the circle be R, then we have AB = 2R sin α, BC = 2R sin β, CD = 2R sin γ, AD = 2R sin (180° − (α + β + γ)), AC = 2R sin (α + β) and BD = 2R 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 4R² has disappeared by dividing both sides of the equation by it.

Now by using the angle 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 cos²β sin γ + cos α sin²β cos γ + cos α sin β cos β sin γ.

Here is another, perhaps more transparent, proof using rudimentary trigonometry. Define a new quadrilateral ABCD' inscribed in the same circle, where A, B, C are the same as in ABCD, and D' located at a new point on the same circle, defined by |AD'| = |CD|, |CD'| = |AD|. Then, ABCD' has the same lengths of edges, and consequently the same inscribed angles subtended by the corresponding edges, as ABCD, only in a different order.

Proof by inversion

Choose an auxiliary circle Γ of radius r centered at D with respect to which the circumcircle of ABCD is inverted into a line. Then A'B' + B'C' = A'C'. Then A'B', B'C' and A'C' can be expressed as AB⋅DB'/DA, BC⋅DB'/DC and AC⋅DC'/DA respectively. Multiplying each term by DA⋅DC/DB' and using DC'/DB' = DB/DC yields Ptolemy's equality.

Proof using complex numbers

Embed ABCD in the complex plane by identifying A ↦ z_A, … , D ↦ z_D as four distinct complex numbers z_A, … , z_D ∈ C. Define the cross-ratio

ζ := (z_A − z_B)(z_C − z_D)/(z_A − z_D)(z_B − z_C) ∈ C ≠ 0.

Then

AB⋅CD + AD⋅BC = |(z_A − z_B)(z_C − z_D)| + |(z_A − z_D)(z_B − z_C)| = (|ζ| + 1)| (z_A − z_D)(z_B − z_C)| ≥ |(ζ + 1)(z_A − z_D)(z_B − z_C)| = |(z_A − z_B)(z_C − z_D) + (z_A − z_D)(z_B − z_C)| = |(z_A − z_C)(z_B − z_D)| = AC⋅BD

with equality if and only if the cross-ratio ζ is a positive real number. This proves Ptolemy's inequality generally, as it remains only to show that z_A, … , z_D lie consecutively arranged on a circle if and only if ζ ∈ R_>0.

Corollaries

When dealing with a circle that has a diameter of one unit, the lengths of the sides of any four-sided shape on the circle match the sine values of certain 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 is a way to understand shapes that are not perfect circles. It tells us that for any four-point shape, a certain math rule will always be true. This rule becomes exactly true only when the four points fit perfectly on a circle, which is when Ptolemy's theorem applies.

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 ratio, or 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 (multiplication result) and the ratio of the diagonals, we can then find the exact lengths of each diagonal.