Question

ABCD is a cyclic parallelogram .show tha it is a rectangle

Answer 1

If ABCD paralelogram is cyclic, then AB and CD are parallel chords of the circle. The orthogonal bisectors of chords go through the center of the circle, so the orthogonal bisector of AB and CD both should go through the center of the circle, but if ABCD is not a rectangle these bisectors are parallel lines and don’t go through the same point.

On the other hand rectangles are always cyclic, the vertices being the same distance from the intersection of the diagonals.

Answer 2

In a cyclic quadrilateral opposite angles are supplementary
Therefore angle A + angle C = 180
But angle A = angle C ( opposite angles of a parallelogram are equal)
Thus 2 angle A =180
angle A =90
Similarly all angles are 90
But ABCD is a parallelogram
Hence ABCD is a rectangle.