Prove that a cyclic parallelogram is a rectangle.

Question

KnowledgeBoat · Accepted Answer

Let ABCD be the cyclic parallelogram. We know that opposite angles of a parallelogram are equal. ⇒ ∠A = ∠C .......(1) ⇒ ∠B = ∠D .......(2) We know that the sum of opposite angles of a cyclic quadrilateral is 180°. ⇒ ∠A + ∠C = 180° ⇒ ∠A + ∠A = 180° (From equation (1)) ⇒ 2∠A = 180° ⇒ ∠A = ⇒ ∠A = 90°. We know that, If one angle of a parallelogram is 90°, then it is a rectangle. Thus, quadrilateral ABCD is a rectangle. Hence, proved that a cyclic parallelogram is a rectangle.

Mathematics

Prove that a cyclic parallelogram is a rectangle.

Circles

Related Questions

If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively. Prove that ∠ACP = ∠QCD.

If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD.