Question

If diagonals of a cyclic quadrilateral are diameters 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 diameters of circle. 

To Prove: ABCD is a rectangle. 

Proof: AC is a diameter. 

∠ABC is angle in semicircle. 

Angle in semicircle is a right angle. 

∴ ∠ABC = 90° 

∠ADC = 90° 

Similarly, BD is a diamgers, 

∠DAB, ∠DCB are angles in semicircle. 

∠DAB = 90° 

∠DCB = 90° 

Now, four angles of quadrilateral ABCD are right angles. 

∴ ∠A = ∠B = ∠C = ∠D = 90° 

∴ ABCD is a rectangle.