Brahmagupta's theorem is a result in geometry. It states that if a cyclic quadrilateral has perpendicular diagonals, then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. It is named after the Indian mathematician Brahmagupta.
More specifically, let be four points on a circle such that the lines are perpendicular. Denote the intersection of by . Drop the perpendicular from to the line , calling the intersection . Let be the intersection of the line and the edge . Then, the theorem states that is in the middle of .
We need to prove that . We will prove that both are in fact equal to .
To prove that , first note that the angles
Hence, is an isosceles triangle, and thus the sides .
The proof that goes similarly. The angles
so is an isosceles triangle, so . It follows that , as the theorem claims.