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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). [1]
Ptolemy's theorem states the relationship between the diagonals and the sides of a cyclic quadrilateral. It is a powerful tool to apply to problems about inscribed quadrilaterals. Let's prove this theorem.
PTOLEMY’S THEOREM AND ITS CONVERSE RICHARD G. SWAN Abstract. This is an expository note on Ptolemy’s Theorem and its converse, giving a more algebraic proof of these results. We show that 4 points in the plane lie on a circle or straight line if and only if they satisfy Ptolemy’s condition. 1. The Theorems
Jan 3, 2017 · The classical theorem of Ptolemy states that if A, B, C, D are, in this order, four points on the circle O, then AC . BD = ABCD . + AD . BC . In the literature of Euclidean geometry there are many proofs for this celebrated theorem (some of them can be found in [3], [4],[7],[8], [9] and [10] ).
Oct 10, 2024 · Ptolemy's Theorem. For a cyclic quadrilateral, the sum of the products of the two pairs of opposite sides equals the product of the diagonals. (1) (Kimberling 1998, p. 223). This fact can be used to derive the trigonometry addition formulas. Furthermore, the special case of the quadrilateral being a rectangle gives the Pythagorean theorem.
Ptolemy's theorem is a powerful result. With its help we establish the Pythagorean Theorem and Carnot's 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.
Aug 9, 2016 · For one thing, Ptolemy's theorem "decays" nicely to $a c = a c$ in the degenerate case where $I \equiv J, b = 0, e = a, f = c$, while similarity-based proofs would not directly translate to the trivial case.
