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

Explanation

We conclude from the given information

`ABCD` is a cyclic quadilateral ,
`AC` and `BD` are diameters of the circle where they meet at center `O` of the circle.

Proof:
In triangle `/_\AOD` and `/_\BOC`,

`OA=OC` (both are radii of same circle)

`/_AOD=/_BOC` (vert.opp∠S)
`OD=OB`(both are radii of same circle)
`:./_\AOD~=/_\BOC`
`=>AD=BC`(C.P.C.T)

Similarly,
by taking `/_\AOB` and `/_\COD,AB=DC`
Also,
`/_BAD=/_ABC=/_BCD=/_ADC=90 ^@`(angle in a semicircle)
`:.ABCD` is a rectangle.