7. 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 = (1/2) NP

and OM = OQ = (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.