# Completing Brahmagupta’s Extension of Ptolemy’s Theorem

Chapter

## 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

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