Cyclic Quadrilateral
A quadrilateral is a 4 sided polygon bounded by 4 finite line segments. The word ‘quadrilateral’ is composed of two Latin words, Quadri meaning ‘four ‘and latus meaning ‘side’. It is a twodimensional figure having four sides (or edges) and four vertices. A circle is the locus of all points in a plane which are equidistant from a fixed point. If all the four vertices of a quadrilateral ABCD lie on the circumference of the circle then ABCD is a cyclic quadrilateral. In other words, if any four points on the circumference of a circle are joined they form vertices of a cyclic quadrilateral. It can be visualized as a quadrilateral which is inscribed in a circle, i.e. all four vertices of the quadrilateral lie on the circumference of the circle.
What is a Cyclic Quadrilateral?
In the figure given below, the quadrilateral ABCD is cyclic.
Let us do an activity. Take a circle and choose any 4 points on the circumference of the circle. Join these points to form a quadrilateral. Now measure the angles formed at the vertices of the cyclic quadrilateral. To our surprise, the sum of the angles formed at the vertices is always 360^{o }and the sum of angles formed at the opposite vertices is always supplementary. This property can be stated as a theorem as:
Cyclic Quadrilateral Theorem
Theorem 1: In a cyclic quadrilateral, the sum of either pair of opposite angles is supplementary.
Proof: Let us now try to prove this theorem.
Given: A cyclic quadrilateral ABCD inscribed in a circle with center O.
Construction: Join the vertices A and C with center O.
The converse of this theorem is also true which states that if opposite angles of a quadrilateral are supplementary then the quadrilateral is cyclic.
Theorem 2: The ratio between the diagonals and the sides is special and is known as Cyclic quadrilateral theorem. If there’s a quadrilateral which is inscribed in a circle, then the product of the diagonals is equal to the sum of the product of its two pairs of opposite sides.
If PQRS is a cyclic quadrilateral, PQ and RS, and QR and PS are opposite sides. PR and QS are the diagonals.
(PQ x RS) + ( QR x PS) = PR x QS
Properties of Cyclic Quadrilateral











Cyclic Quadrilateral Examples
Question: Find the value of angle D of a cyclic quadrilateral, if angle B is 60^{o}?
Solution:
As ABCD is a cyclic quadrilateral, so the sum of a pair of two opposite angles will be 180^{o}.
∠B + ∠ D = 180^{o}
600 + ∠D = 180^{o}
∠D = 1800 – 60^{o}
∠D = 120^{o}
The value of angle D is 120^{o}.
Question: Find the value of angle D of a cyclic quadrilateral, if angle B is 80^{o}?
Solution:
As ABCD is a cyclic quadrilateral, so the sum of a pair of two opposite angles will be 180^{o}.
∠B + ∠ D = 180^{o}
800 + ∠D = 180^{o}
∠D = 1800 – 80^{o}
∠D = 100^{o}
The value of angle D is 100^{o}.