# Brahmagupta-Mahavira Identities

In a cyclic quadrilateral $ABCD$ with sides $a,$ $b,$ $c,$ $d,$ as shown, the diagonals can be computed via

$\begin{align}\displaystyle m^{2} &= \frac{(ab + cd)(ac + bd)}{ad + bc}\\ n^{2} &= \frac{(ac + bd)(ad + bc)}{ab + cd}. \end{align}$

Most of the sources attribute this result to the great 9^{th} century Indian mathematician Mahavira (or Mahaviracharya, meaning Mahavira the Teacher). However, according to Richard Askey with a reference to Henry Thomas Colebrooke the formulas have been known to another great Indian mathematician Brahmagupta already in the 7^{th} century.

We have established these identities elsewhere in two ways. The gem of a proof that follows is due to yet another Indian mathematician Paramesvara who worked in the 15^{th} century. This is really a beautiful

### Proof

Ptolemy's formula in a cyclic quadrilateral tells us that

$n\cdot m = b\cdot d + a\cdot c.$

Let's interchange the sides $a$ and $d:$

The operation will leave the quadrilateral cyclic and the diagonal $m$ unchanged. If the other diagonal is $u,$ the Ptolemy's formula gives,

$mu = a\cdot b + c\cdot d.$

Similarly, an exchange of $a$ and $b$ yields a cyclic quadrilateral with diagonals $u$ and $n;$ we have the identity:

$un = a\cdot d + b\cdot c.$

We get $m^{2}$ multiplying the first two and dividing by the third identity:

$\displaystyle m^{2} = \frac{(nm)(mu)}{un} = \frac{(b\cdot d + a\cdot c)(a\cdot b + c\cdot d)}{a\cdot d + b\cdot c}.$

For $n^{2},$ we have,

$\displaystyle n^{2} = \frac{(nm)(un)}{mu} = \frac{(b\cdot d + a\cdot c)(a\cdot d + b\cdot c)}{a\cdot b + c\cdot d}.$

And, for the ratio $\displaystyle\frac{m}{n},$ *Ptolemy's second theorem*

$\displaystyle\frac{m}{n} = \frac{a\cdot b + c\cdot d}{a\cdot d + b\cdot c}.$

### References

- R. Askey,
__Completing Brahmagupta's Extension of Ptolemy's Theorem__, in K. Alladi et al. (eds.),*The Legacy of Alladi Ramakrishnan in the Mathematical Sciences*, Springer Science+Business Media, LLC 2010 - H. T. Colebrooke,
*Algebra: With Arithmetic and Mensuration From The Sandskrit of Brahmagupta and Bhascara*, 1817, reprinted, Kessinger, Whitefish, MT, USA, - J. L. Heilbron,
*Geometry Civilized*, Oxford University Press, Oxford, 2000, p. 219.

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