Question

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

Answer 1

Data: In cyclic quadrilateral $ABCD,AC$ and $BD$ are diameter of circle
To prove : $ABCD$ is a rectangle.
Proof: $AC$ is a diameter. $\angle ABC$ is angle in semicircle. Angle in semicircle is a right angle.
$therefore\angle ABC={90}^o\angle ADC={90}^o$
Similarly, $BD$ is a diameter, $\angle DAB,\angle DCB$ are angles in semicircle
$\angle DAB={90}^o\angle DCB={90}^o$
Now, four angles of quadrilateral $ABCD$ are right angles.
$\therefore \angle A=\angle B=\angle C=\angle D={90}^o$
$\therefore ABCD$ is a rectangle.