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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

Timeline

PythagoreanTheorem timeline Babylonian Mud Wall Tablet Babylonian Inner Diagonal Tablet Babylonian Square Root of 2 Tablet Berlin Pythagorean Theorem Papyrus Plimpton 322 Vedic Fire Altar Rectangle Diagonal Vedic Fire Altar Square Diagonal Euclid's Elements Ptolemy's Quadrilateral Theorem Zhoubi Suanjing Pythagorean Theorem Proof Ibn Qurra's Pythagorean Theorem Proofs Lilāvatī of Bhāskara II

Interactive Content

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.