If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it - Maths - Circles

Question

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

Gopal Mohanty · Accepted Answer

Let ABCD be a cyclic quadrilateral having diagonals BD and AC, intersecting each other at point O

(Consider BD as a chord) ∠BCD + ∠BAD = 180° (Cyclic quadrilateral) ∠BCD = 180° − 90° = 90°

(Considering AC as a chord) ∠ADC + ∠ABC = 180° (Cyclic quadrilateral) 90° + ∠ABC = 180° ∠ABC = 90° Each interior angle of a cyclic quadrilateral is of 90° Hence, it is a rectangle

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