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 , , , , , and , so the general result is written
(2)
For a rectangle , , , and , so the theorem gives
(3)
See also Concyclic ,
Cyclic Quadrilateral ,
Fuhrmann's Theorem ,
Ptolemy
Inequality ,
Pythagorean Theorem ,
Quadrilateral ,
Tweedie's Theorem
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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. New York: Chelsea, p. 38,
1971. Coxeter, H. S. M. and Greitzer, S. L. Geometry
Revisited. Washington, DC: Math. Assoc. Amer., pp. 42-43, 1967. Durell,
C. V. Modern
Geometry: The Straight Line and Circle. London: Macmillan, p. 17, 1928. Johnson,
R. A. "The Theorem of Ptolemy." §92 in Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.
Boston, MA: Houghton Mifflin, pp. 62-63, 1929. Kimberling, C. "Triangle
Centers and Central Triangles." Congr. Numer. 129 , 1-295, 1998. Wells,
D. The
Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 200-201, 1991. Referenced on Wolfram|Alpha Ptolemy's Theorem
Cite this as:
Weisstein, Eric W. "Ptolemy's Theorem."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/PtolemysTheorem.html
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