Ptolemy's theorem

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).

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: 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:[4] 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 [6] is thereafter calculated as As Copernicus (following Ptolemy) wrote, The animation here shows a visual demonstration of Ptolemy's theorem, based on Derrick & Herstein (2012).

[9] 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.

centered at D with respect to which the circumcircle of ABCD is inverted into a line (see figure).

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.

This proves Ptolemy's inequality generally, as it remains only to show that

lie consecutively arranged on a circle (possibly of infinite radius, i.e. a line) in

Note that this proof is equivalently made by observing that the cyclicity of ABCD, i.e. the supplementarity

is 0 (i.e. all three products are positive real numbers), and by which Ptolemy's theorem is then directly established from the simple algebraic identity 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

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

it is possible to derive a number of important corollaries using the above as our starting point.

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.

[11] This derivation corresponds to the Third Theorem as chronicled by Copernicus following Ptolemy in Almagest.

This was a critical step in the ancient method of calculating tables of chords.

[12] This corollary is the core of the Fifth Theorem as chronicled by Copernicus following Ptolemy in Almagest.

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 (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.

The equation in Ptolemy's theorem is never true with non-cyclic quadrilaterals.

Writing the area of the quadrilateral as sum of two triangles sharing the same circumscribing circle, we obtain two relations for each decomposition.

Consequence: Knowing both the product and the ratio of the diagonals, we deduce their immediate expressions: