What is Cyclic Quadrilateral?
If the four vertices of a quadrilateral lie on the circumference of the circle then it is known as a cyclic quadrilateral.
A cyclic quadrilateral is also known as a circumcircle or a circumscribed cycle.
The vertices of a quadrilateral are said to be concyclic.
Examples of cyclic quadrilaterals.
What is a Quadrilateral?
A quadrilateral is a polygon having four edges and four corners or vertices.
The sum of interior angles of a quadrilateral sum up to 360 degrees.
The word ‘quadrilateral’ is composed of two Latin words, Quadri which means ‘four’ and latus which means ‘side’.
A quadrilateral is a twodimensional figure having four edges.
Properties of Cyclic Quadrilateral –
Here’s a table that lists down the properties of a cyclic quadrilateral.
Formula for the Perimeter of a Cyclic Quadrilateral 
Formula for the Area of an Inscribed or Cyclic Quadrilateral 
Cyclic Quadrilateral Theorem
Here is the important cyclic quadrilateral theorem
Theorem of Cyclic Quadrilateral (I)
The either pair of the opposite angles of a cyclic quadrilateral sum up to 180° .
Given, ABCD is a cyclic quadrilateral of a cycle with the centre as O.
Here, \[\angle BAD + \angle BCD = {180^o}\]
\[\angle ABC + \angle ADC = {180^o}\]
Therefore, cyclic quadrilateral angles equal to 180 degrees.
Theorem of Cyclic Quadrilateral (II)
In a cyclic quadrilateral, if a quadrilateral is inscribed inside a cycle, the product of the diagonals of the cyclic quadrilateral is equal to the sum of the two pairs of opposite sides of the cyclic quadrilateral.
In a cyclic quadrilateral ABCD, AC and BD are diagonals and AB, CD, AD and BC are opposite sides.
Product of Diagonals 
\[\left( {AC \times BD} \right){\text{ }} = {\text{ }}\left( {AB \times CD} \right){\text{ }} + \left( {AD \times BC} \right)\]
Ratio of Diagonals 
\[\frac{{AC}}{{BD}} = \frac{{\left( {AB \times AD} \right) + \left( {BC \times CD} \right)}}{{\left( {AB \times BC} \right) + \left( {AD \times CD} \right)}}\]
Questions on Cyclic Quadrilateral Angles and Based on Cyclic Quadrilateral Theorem
Question 1: What will be the value of angle B of a cyclic quadrilateral if the value of angle D is equal to 60 degrees.
Solution: Let’s list down the given information, \[\angle B\] = 60°
As quadrilateral ABCD is cyclic, which means that the sum of a pair of two opposite angles in a cyclic quadrilateral will be equal to 180° according to the cyclic quadrilateral theorem.
\[\angle B\]+\[ \angle D\] = 180^{0}
60^{0}+ \[\angle D\] = 180^{0}
\[\angle D\] = 180^{0}  60^{0}
Therefore, the value of\[\angle D\] = 120^{0}
Question 2: What will be the value of angle D of a cyclic quadrilateral if the value of angle B is equal to 70 degrees.
Solution: Let’s list down the given information, \[\angle D = {70^o}\]
As quadrilateral ABCD is cyclic, which means that sum of a pair of two opposite angles in a cyclic quadrilateral will be equal to 180° according to the cyclic quadrilateral theorem.
\[\angle B\] + \[\angle D\] = 180^{0}
\[\angle B\] + 70^{0} = 180^{0}
\[\angle B\] = 180^{0}  70^{0}
Therefore, the value of\[\angle D = {110^o}\]
Question 3: Find the perimeter of a cyclic quadrilateral with sides 4cm, 2 cm, 6 cm and 8 cm.
Solution: Given the measurement of the sides are,
4cm, 2 cm, 6 cm and 8 cm
Using the formula of perimeter,
Perimeter = 2s
\[S = \frac{{a + b + c + d}}{2}\]
\[S = \frac{{4 + 6 + 2 + 8}}{2}\]
s= 10
Therefore, perimeter of a cyclic quadrilateral = 2s = 20
Question 1) What are the Properties of Cyclic Quadrilaterals?
Answer)








AE × EC = BE × ED 
Question 2) What is a Cyclic Quadrilateral?
Answer) A foursided geometric figure whose vertices lie on the circumference of the circle is known as a cyclic quadrilateral.
Question 3) Are all Rectangles Cyclic?
Answer) Yes, all rectangles are cyclic, but many quadrilaterals are not cyclic. As we know that if the opposite angles of a quadrilateral are supplementary which means that the sum of either pair of opposite angles is equal to 180 degrees only then it is cyclic.