Ptolemy of Alexandria (~100-168) gave the name to the Ptolemy's Planetary theory which he
described in his treatise Almagest. The book is mostly devoted to astronomy and trigonometry where, among
many other things, he also gives the approximate value of π as 377/120 and proves the theorem that now bears his name. The name Almagest is actually a corruption of the Arabic rendition "Al Magiste" - The Greatest - of the Greek H Megisth Suntaxiz (E Megiste Syntaxis).
This classical theorem has been proved many times over. Following is the simplest proof
I am aware of. (There is another simple proof of a recent vintage.)
On the diagonal BD locate a point M such that angles ACB and MCD be equal. Since angles BAC and BDC subtend the same arc, they are equal. Therefore, triangles ABC and DMC are similar. Thus we get CD/MD = AC/AB, or AB·CD = AC·MD.
Now, angles BCM and ACD are also equal; so triangles BCM and ACD are similar which leads to BC/BM = AC/AD, or BC·AD = AC·BM. Summing up the two identities we obtain
The following problem is discussed in Honsberger, Mathematical Morsels, p172:
Let A1A2A3 denote an equilateral triangle inscribed in a circle.
For any point P on the circle, show that the two shorter segments among PA1, PA2, PA3
add up to the third one.
Solution
Let s denote the length of the side of the given triangle. By Ptolemey's Theorem we have
s·PA1 = s·PA2 + s·PA3
Therefore,
PA1 = PA2 + PA3
Remark
This result has an interesting generalization to the case of a regular 3n-gon inscribed
in a circle: Of the 3n chords obtained by connecting a point P with vertices of the polygon,
the sum of the 2n shortest ones equals the sum of the n longest.
The problem itself is sometimes attributed to Van Schooten (1615 - 1660), see, for example, The Changing Shape of Geometry, (C. Pritchard, Cambridge University Press, 2003), p. 184, where two additional proofs could be found.
Remark
Ptolemy's theorem is a powerful result. With its help we establish the Pythagorean Theorem. Combined with the Law of Sines, Ptolemy's theorem serves to prove the addition and subtraction formulas for the sine function. It has a short proof in complex numbers. The following generalization is sometimes attributed to the great 9th century Indian mathematician Mahavira (or Mahaviracharya, meaning Mahavira the Teacher). However, the formulas derived below have been already known to the 7th Indian mathematician Brahmagupta.
In a cyclic quadrilateral ABCD, let a, b, c, d denote the lengths of sides AB, BC, CD, DA, and m, n the lengths of the diagonals BD and BC. Then Mahavira's result is expressed as
m² = (ab + cd)(ac + bd)/(ad + bc) and
n² = (ac + bd)(ad + bc)/(ab + cd)
H. Eves gives a proof as a sequence of exercises in [Great Moments in Mathematics Before 1650, p. 108]:
Let t be the diameter of the circmcircle of ABCD and θ the angle between either diagonal and the perpendicular upon the other.
Then, (using triangle's formula ab = 2hR applied to DAB and DCB), we get
mt·cosθ = ab + cd and
nt·cosθ = ad + bc
So m / n = (ab + cd) / (ad + bc) which is called Ptolemy's second theorem.
Also mn = ac + bd (Ptolemy relation)
Multiplying those last 2 equations, we get:
m² = (ab + cd)(ac + bd) / (ad + bc)
Dividing instead, we get:
n² = (ac + bd)(ad + bc) / (ab + cd).
Finally, we also get
(t cosθ)² = (ab + cd)(ad + bc) / (ac + bd).
And if the diagonals in the quadrilateral are orthogonal,
t² = (ab + cd)(ad + bc) / (ac + bd).
An additional derivation has been posted to the CTK Exchange. This one can be found in Advanced Trigonomentry by C. V. Durrell and A. Robson, 1930, p. 25. (The book is available in a 2003 Dover edition and on google's bookshelf.)
In the same notations as above, by the Cosine Rule
