Cyclic Quadrilateral
A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It is also sometimes called inscribed quadrilateral. The circle which consists of all the vertices of any polygon on its circumference is known as the circumcircle or circumscribed circle.
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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 the 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.
Cyclic Quadrilateral Definition
The definition states that a quadrilateral which is circumscribed in a circle is called a cyclic quadrilateral. It means that all the four vertices of quadrilateral lie in the circumference of the circle. Let us understand with a diagram.
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. It is noted that 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.
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Cyclic Quadrilateral Angles
The sum of the opposite angles of a cyclic quadrilateral is supplementary.
Let ∠A, ∠B, ∠C and ∠D are the four angles of an inscribed quadrilateral. Then,
∠A + ∠C = 180°
∠B + ∠D = 180°
Therefore, an inscribed quadrilateral also meets the angle sum property of a quadrilateral, according to which, the sum of all the angles equals 360 degrees. Hence,
Radius of Cyclic Quadrilateral
If a, b, c and d are the successive sides of a cyclic quadrilateral, and s is the semi perimeter, then the radius is given by,
\(R = \frac{1}{4}\sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(sa)(sb)(sc)(sd)}}\)Diagonals of Cyclic Quadrilaterals
Suppose a,b,c and d are the sides of a cyclic quadrilateral and p & q are the diagonals, then we can find the diagonals of it using the below given formulas:
\(p=\sqrt{\frac{(a c+b d)(a d+b c)}{a b+c d}} \text { and } q=\sqrt{\frac{(a c+b d)(a b+c d)}{a d+b c}}\)
Area of Cyclic Quadrilateral
If a,b,c and d are the sides of a inscribed quadrialteral, then its area is given by:
\(Area=\sqrt{(sa)(sb)(sc)(sd)}\)
Where s is the semiperimeter.
Cyclic Quadrilateral Theorems
There are two important theorems which prove the cyclic quadrilateral.
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 can be defined 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











Problems and Solutions
Question: Find the value of angle D of a cyclic quadrilateral, if angle B is 60^{o}.
Solution:
If ABCD is a cyclic quadrilateral, so the sum of a pair of two opposite angles will be 180°.
∠B + ∠ D = 180°
60° + ∠D = 180°
∠D = 180° – 60°
∠D = 120°
The value of angle D is 120°.
Question: Find the value of angle D of a cyclic quadrilateral, if angle B is 80°.
Solution:
If ABCD is a cyclic quadrilateral, so the sum of a pair of two opposite angles will be 180°.
∠B + ∠ D = 180°
80° + ∠D = 180°
∠D = 180° – 80°
∠D = 100°
The value of angle D is 100°.
You should practise more examples using cyclic quadrilateral formulas to understand the concept better.
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