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Brahmagupta's Problem

Brahmagupta was a Hindu mathematician of the seventh century AD who discovered a neat formula for the area of a cyclic quadrilateral.

Cyclic quad

If the side lengths are a, b, c and d, and s=(a+b+c+d)/2 (s is the semi-perimeter of the quadrilateral), then the area is given by the formula:


The proof of Brahmaghupta's formula requires a good deal of trigonometry embedded in rather a lot of crunchy algebra, so we'll leave it for another day. But it is possible to prove that if a cyclic quadrilateral has perpendicular diagonals crossing at P, the line through P perpendicular to any side bisects the opposite side.