**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 .

## Proof

We need to prove that . We will prove that both are in fact equal to .

To prove that , first note that the angles

because they are inscribed angles that intercept the same arc of the circle. Furthermore, the angles are both complementary to angle (i.e., they add up to 90°). Finally, 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.

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