# 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

### Statement

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

Proposition: Let ${\displaystyle ABCD}$ is a quadrilateral inscribed in a circle with perpendicular diagonals ${\displaystyle AC}$ and ${\displaystyle BD}$ intersecting at point ${\displaystyle M}$. ${\displaystyle ME}$ is a perpendicular on the side ${\displaystyle BC}$ from the point ${\displaystyle M}$ and extended ${\displaystyle EM}$ intersects the opposite side ${\displaystyle AD}$ at point ${\displaystyle F}$. It is to be proved that ${\displaystyle AF=DF}$.

Proof: ${\displaystyle \angle CBD=\angle CAD}$ [As both are inscribed angles that intercept the same arc ${\displaystyle CD}$ of a circle]

Or, ${\displaystyle \angle CBM=\angle MAF}$

Here, ${\displaystyle \angle CMB+\angle CBM+\angle BCM=180}$°

Or, ${\displaystyle \angle CMB+\angle BCM=180}$° ${\displaystyle -\angle CBM}$

Again, ${\displaystyle \angle CME+\angle CEM+\angle ECM=180}$°

Or, ${\displaystyle \angle CME+\angle CMB+\angle BCM=180}$° [As ${\displaystyle \angle CMB=\angle CEM=90}$° and ${\displaystyle \angle BCM=\angle ECM}$]

Or, ${\displaystyle \angle CME+180}$° ${\displaystyle -\angle CBM=180}$°

Or, ${\displaystyle \angle CME=\angle CBM}$

Or, ${\displaystyle \angle AMF=\angle MAF}$ [As, \angle AMF = \angle CME; Vertical Angles]

Therefore, ${\displaystyle AF=MF}$

In the similar way, ${\displaystyle \angle MDF=\angle DMF}$ and ${\displaystyle MF=DF}$

Or, ${\displaystyle AF=DF}$ [Proved]

## Reference

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