# Completing Brahmagupta’s Extension of Ptolemy’s Theorem

Chapter

First Online:

## Summary

Brahmagupta extended Ptolemy’s theorem on cyclic quadrilaterals to find the lengths of the diagonals, the segments made when they are cut at the point of intersection of the diagonals, and the lengths of the sides of the needles, the figures formed when opposite sides of the quadrilateral are extended until they meet. Proofs of these results are given, and a derivation of the 19th century result of the length of the third diagonal is given. This “diagonal” is formed by connecting the tips of the needles with a line segment.

### Key words and phrases

Ptolemy’s theorem Brahmagupta Third diagonal of cyclic quadrilateral### References

- 1.Bogomolny, Alexander,
*Ptolemy’s Theorem*. http://www.cut-the-knot.org/proofs/ptolemy.shtml. - 2.Colebrooke, Henry Thomas,
*Algebra: With Arithmetic and Mensuration From The Sandskrit of Brahmagupta and Bhascara*, 1817, reprinted, Kessinger, Whitefish, MT, USA, 2008.Google Scholar - 3.Crilly, Tony and Colin Fletcher,
*Ptolemy’s Theorem, Its Parent and Offspring*, in [10, pp. 42-49].Google Scholar - 4.Durell, C.V. and A. Robson,
*Advanced Trigonometry*, G. Bell, London, 1930, reprint, Dover. Mineola, N.Y., 2003.MATHGoogle Scholar - 5.Givental, A., translator and editor,
*Kiselev’s Geometry, Book 1*, Planimetry, Sumizdat, El Cerrito, CA, 2006.Google Scholar - 6.Heath, Thomas L.,
*Euclid’s Elements, Vol. 2*, second edition, Cambridge Univ. Press, 1926, reprinted, Dover, New York, 1956.MATHGoogle Scholar - 7.Heilbron, J.L.,
*Geometry Civilized*, Clarendon Press, Oxford, 1998.MATHGoogle Scholar - 8.Hobson, E. W.,
*A Treatise on Plane and Advanced Trigonometry*, Cambridge Univ. Press, Cambridge, first edition, 1891, seventh edition, 1928, reprinted, Dover, Mineola, NY, 2005.Google Scholar - 9.McDowell, J.,
*Exercises on Euclid and in Modern Geometry*, Deighton Bell, Cambridge, 1878, available through google book search.Google Scholar - 10.Pritchard, Chris,
*The Changing Shape of Geometry*, Cambridge Univ. Press, Cambridge, 2003.MATHGoogle Scholar - 11.Toomer, G. J., translator,
*Ptolemy’s Almagest*, Dukworth, London, 1984, reprinted Princeton Univ. Press, princeton, 1998.Google Scholar

## Copyright information

© Springer New York 2010