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around 150 CE

Ptolemy's Quadrilateral Theorem

The Pythagorean theorem via cyclic quadrilaterals

Claudius Ptolemy's famous work the Almagest gives a result on cyclic quadrilaterals, which gives the Pythagorean theorem as a special case.

Ptolemy's Quadrilateral Theorem

Book 1, Chapter 10 of the Almagest contains a relation between the sides and diagonals of a cyclic quadrilateral known as Ptolemy's theorem: AB×CD + BC×DA = AC×BD. Taking the special case of the quadrilateral being a rectangle, this result reduces to the Pythagorean theorem, thus providing one of the world's first alternate proofs.

Artifact format

Unknown (lost)

Artifact origin

Alexandria, Egypt


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Computational Explanation

Other Resources

Additional Reading

  • Coolidge, J. L. "A Historically Interesting Formula for the Area of a Quadrilateral." Amer. Math. Monthly, Vol. 46, pp. 345–347, 1939.
  • Coxeter, H. S. M. and Greitzer, S. L. "Ptolemy's Theorem and Its Extensions." §2.6 in Geometry Revisited. Washington, DC: Mathematical Association of America, pp. 42–43, 1967.
  • Durrell, C. V. and Robson, A. Advanced Trigonometry. p. 25, 1930.
  • 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., No. 129, p. 223, 1998.
  • Maor, E. The Pythagorean Theorem: A 4000-Year History. Princeton, NJ: Princeton University Press, pp. 104–105, 2007.
  • Ptolemy, C. Almagestum. Latin translation of Gerard of Cremona. Venice, Italy: Petrus Lichtenstein, left leaf of p. 6, 1515.
  • Toomer, G. J. Ptolemy's Almagest. Princeton, NJ: Princeton University Press, 1988.
  • Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 200–201, 1991.