# Timeless Theorems of Mathematics/Brahmagupta Theorem

The Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.^{[1]}

The theorem is named after the Indian mathematician Brahmagupta (598-668).

## Proof[edit | edit source]

### Statement[edit | edit source]

If any cyclic quadrilateral has perpendicular diagonals, then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.

### Proof[edit | edit source]

**Proposition:** Let is a quadrilateral inscribed in a circle with perpendicular diagonals and intersecting at point . is a perpendicular on the side from the point and extended intersects the opposite side at point . It is to be proved that .

**Proof:** [As both are inscribed angles that intercept the same arc of a circle]

Or,

Here, °

Or, °

Again, °

Or, ° [As ° and ]

Or, ° °

Or,

Or, [As, \angle AMF = \angle CME; Vertical Angles]

Therefore,

In the similar way, and

Or, **[Proved]**

## Reference[edit | edit source]

- ↑ Michael John Bradley (2006). The Birth of Mathematics: Ancient Times to 1300. Publisher Infobase Publishing. ISBN 0816054231. Page 70, 85.