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

Statement

edit

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
 
If   and  then   according to the Brahmagupta's theorem

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