If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

Given:
Diagonals NP and QM of a cyclic quadrilateral are diameters of the circle through the vertices M, P, Q and N of the quadrilateral NQPM.



To prove: Quadrilateral NQPM is a rectangle.

Proof:
ON = OP = OQ = OM ...(Radii of circle)

Now, ON = OP = (\(\frac{1}{2} \) ) NP
and OM = OQ = (\(\frac{1}{2} \) ) MQ
\(\therefore \) NP = MQ

Hence, the diagonals MPQN are equal and bisect each other.

Hence, it is proved that quadrilateral NQPM is a rectangle.