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Ptolemy's Theorem

PtolemysTheorem

For a cyclic quadrilateral, the sum of the products of the two pairs of opposite sides equals the product of the diagonals

AB×CD+BC×DA=AC×BD
(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 a=AB, b=BC, c=CD, d=DA, p=AC, and q=BD, so the general result is written

ac+bd=pq.
(2)

For a rectangle, c=a, d=b, and p=q, so the theorem gives

a^2+b^2=p^2.
(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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