Ptolemy's Theorem

In Trigonometric Delights (Chapter 6), Eli Maor discusses this delightful theorem that is so useful in trigonometry. In case you cannot get a copy of his book, a proof of the theorem and some of its applications are given here. The theorem refers to a quadrilateral inscribed in a circle. As you know, three points determine a circle, so the fourth vertex of the quadrilateral is constrained, and the quadrilateral is not a general one. This constraint gives it the property that the product of the diagonals is the sum of the products of opposite sides

Refer to the diagram at the left. The sides of the quadrilateral are chords of the circle, so the angles that each subtends at points on the circumference (such as A, B, C, and D) are equal. The angles marked with single and triple arcs are equal for this reason. Ptolemy's proof uses the line BE drawn so that the angles marked with double arcs are equal. The point E divides the diagonal AC so that AE + EC = AC. The triangles ABE and DBC are similar, because their angles are equal. Since sides in similar triangles are proportional, AE/DC = AB/BD. The triangles CBE and DBA are also similar, so that EC/AD = BC/BD. These ratios are not easy to see; if you have difficulty, draw the triangles separately. From these two equalities, we have AE + EC = AB.DC/BD + AD.BC/BD = AC, from which the theorem follows on multiplication by BD. This theorem is not in Euclid; the first we hear of it is in the Almagest, but it may have been known to Apollonius much earlier.

If the quadrilateral is a rectangle, the Pythagorean theorem follows at once, because the opposite sides are the sides of right triangles, and the diagonals, which are diameters of the circle, are the hypotenuses. Sketch the diagram and verify this for yourself. If the circle is assumed to have unit diameter, then the chords are equal to the sines of the angles they subtend at the circumference. By taking special quadrilaterals, important trigonometric identities are obtained. For example, if one diagonal is a diameter, then it subtends right angles at the circumference, so that the sides of the quadrilateral are the sines and cosines of the two acute angles of the right triangles, and the other diagonal is the sine of the sum of the acute angles (draw a diagram!). Hence, we get sin(α+β) = sin α cos β + cos α sin β. The formula for the sine of the difference of two angles can be obtained in the same way by taking the other two vertices of the quadrilateral on the same side of the diameter, instead of on opposite sides.

Ptolemy used results like these to calculate his tables of sines. When a triangle is inscribed in a circle of unit diameter, the sides are equal to the sines of the angles opposite them (this is a consequence of, or a reason for, the Law of Sines. Can you prove this? Use trigonometry for an easy proof.). Ptolemy's tables give the chords, not the half-chords that correspond to angles at the center of the circle, as is the current practice. Claudius Ptolemaeus was not of the old Greek ruling family of Egypt (Cleopatra was the last of these); he just has the same name, and flourished about a century after Cleo, who was a nasty bit of work.

Composed by J. B. Calvert.
Last revised 14 June 1999.