Ptolemy's theorem states that 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 . In particular, let
(2)
For a rectangle ,
(3)
A diagram of the theorem appears in the 1515 edition
of Ptolemy's Almagest shown below (History of Mathematics Project).
See also Concyclic ,
Cyclic Quadrilateral ,
Fuhrmann's Theorem ,
Ptolemy
Inequality ,
Pythagorean Theorem ,
Quadrilateral ,
Tweedie's Theorem
Explore with Wolfram|Alpha
References Coolidge, J. L. "A Historically Interesting Formula for the Area of a Quadrilateral." Amer. Math. Monthly 46 , 345-347,
1939. Coolidge, J. L. A
Treatise on the Geometry of the Circle and Sphere. Coxeter, H. S. M. and Greitzer, S. L. Geometry
Revisited. Durell,
C. V. Modern
Geometry: The Straight Line and Circle. History of Mathematics Project. "Ptolemy's Quadrilateral Theorem."
Image: University of Vienna, 1515 Almagestum , p. 6, public domain. https://www.history-of-mathematics.org/artifacts/ptolemys-quadrilateral-theorem . Johnson,
R. A. "The Theorem of Ptolemy." §92 in Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Kimberling, C. "Triangle
Centers and Central Triangles." Congr. Numer. 129 , 1-295, 1998. Mathologer.
"Ptolemy's Theorem and the Almagest: We Just Found the Best Visual Proof in
2000 Years." Sept. 7, 2024. https://www.youtube.com/watch?v=rr1fzjvqztY . Wells,
D. The
Penguin Dictionary of Curious and Interesting Geometry. Referenced on Wolfram|Alpha Ptolemy's Theorem
Cite this as:
Weisstein, Eric W. "Ptolemy's Theorem."
From MathWorld https://mathworld.wolfram.com/PtolemysTheorem.html
Subject classifications
, 
, 
, 
, 
, and 
, so the general result is written
, 
, and 
, so the theorem gives
