A quadrilateral is called Cyclic Quadrilaterals if its all vertices lie on a circle.
It has some special properties which other quadrilaterals, in general, need not have. We shall state and prove these properties as theorems. They are as follows :
1) The sum of either pair of opposite angles of a cyclic- quadrilateral is 1800
The opposite angles of cyclic quadrilateral are supplementary.
∠A + ∠C = 1800 and ∠B + ∠D = 1800
Converse of the above theorem is also true.
If the opposite angles are supplementary then the quadrilateral is a cyclic-quadrilateral.
2) If one side of a cyclic-quadrilateral is produced, then the exterior angle is equal to the interior opposite angle.
ABCD is a cyclic-quadrilateral then ∠CBE = ∠ADC
3) If two non-parallel sides of trapezoid ( trapezium ) are equal, it is cyclic.
If AD = CB then the trapezoid ABCD is a cyclic-quadrilateral. Some solved examples on the above results
1) ABCD is a cyclic-quadrilateral; O is the center of the circle. If ∠BOD = 1600, find the measure of ∠BPD and ∠BCD.
∠BOD = 1600 ( given and it’s a central angle )
∴ ∠BAD = ½ ∠BOD = 800
As ABCD is a cyclic-quadrilateral,
⇒ ∠BAD + ∠BPD = 1800
80 + ∠BPD = 180
⇒ ∠BPD = 1000
⇒ ∠BCD = 1000 ( Angles in the same segment )
2) ABCD is a cyclic-quadrilateral. AB and DC are produced to meet in E. Prove that ΔEBC ~ ΔEDA. Solution : Given : ABCD is a cyclic-quadrilateral.
Prove that : ΔEBC ~ ΔEDA.
1) ABCD is cyclic-quadrilateral.
2) ∠EBC = ∠EDA
2) Exterior angle in a cyclic-quadrilateral is equal to the opposite interior angle.
3) ∠ECB = ∠EAD
3) Exterior angle in a cyclic-quadrilateral is equal to the opposite interior angle.