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 consist of all the vertices of any polygon on its circumference is known as the circumcircle or circumscribed circle

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 two-dimensional 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.

Cyclic Quadrilateral Definition

The definition states that a quadrilateral which 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.

Cyclic Quadrilateral 1

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 360o 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 meet the angle sum property of a quadrilateral, according to which, the sum of all the angles equals 360 degrees. Hence,

∠A + ∠B + ∠C + ∠D= 360° 

Radius of Cyclic Quadrilateral

The circumradius of cyclic quadrilateral is expressed in terms of the two diagonals and distance between the midpoints of the diagonals.

Let p and q are the diagonals of cyclic quadrilateral and x is the distance between the midpoints of diagonals p and q. Then the radius of this quadrilateral will be:

\(R=\sqrt{\frac{p^{2}+q^{2}+4 x^{2}}{8}}\)

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:

\(Area=\sqrt{(s-a)(s-b)(s-c)(s-d)}\)

Where s is the semiperimeter.

s = ½(a+b+c+d)

Cyclic Quadrilateral Theorems

There is 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.

Cyclic Quadrilateral 2

Cyclic Quadrilateral Theorems

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
Cyclic Quadrilateral 3

Properties of Cyclic Quadrilateral

  • In a cyclic quadrilateral, the sum of a pair of opposite angles is 1800(supplementary)
  • If the sum of two opposite angles are supplementary then it’s a cyclic quadrilateral
  • The area of a cyclic quadrilateral is [s(s-a)(s-b)(s-c)(s-c)]0.5 where a, b, c, and d are the four sides of the quadrilateral and the perimeter is 2s
  • The four vertices of a cyclic quadrilateral lie on the circumference of the circle
  • To get a rectangle or a parallelogram, just join the midpoints of the four sides in order
  • If PQRS is a cyclic quadrilateral, then ∠SPR = ∠SQR, ∠QPR = ∠QSR, ∠PQS = ∠PRS, ∠QRP = ∠QSP.
  • If T is the point of intersection of the two diagonals, PT X TR = QT X TS
  • The exterior angle formed if any one side of the cyclic quadrilateral is produced is equal to the interior angle opposite to it
  • In a given cyclic quadrilateral, d1 / d2  = sum of the product of opposite sides, which shares the diagonals endpoints
  • If it is cyclic quadrilateral then the perpendicular bisectors will be concurrent compulsorily
  • In a cyclic quadrilateral, the four perpendicular bisectors of the given four sides meet at the centre O

Problems and Solutions

Question: Find the value of angle D of a cyclic quadrilateral, if angle B is 60o.

Solution:

As ABCD is a cyclic quadrilateral, so the sum of a pair of two opposite angles will be 180o.

∠B + ∠ D = 180o  

600 + ∠D = 180o

∠D = 180o – 60o

∠D = 120o

The value of angle D is 120o.

Question: Find the value of angle D of a cyclic quadrilateral, if angle B is 80o.

Solution:

As ABCD is a cyclic quadrilateral, so the sum of a pair of two opposite angles will be 180o.

∠B + ∠ D = 180o 

80o + ∠D = 180o

∠D = 180o – 80o

∠D = 100o

The value of angle D is 100o.

You should practice more examples using cyclic quadrilateral formulas to understand the concept better.

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Frequently Asked Questions – FAQs

What is a cyclic quadrilateral?

The quadrilateral whose vertices lies on the circumference of a circle is a cyclic quadrilateral. It is also called as an inscribed quadrilateral.

What is the property of cyclic quadrilateral?

The opposite pairs of angles are supplementary to each other.

What is the sum of opposite angles of cyclic quadrilateral?

The sum of opposite angles of cyclic quadrilateral equals 180 degrees.

Is square a cyclic quadrilateral?

Yes, we can draw a cyclic square, whose all four vertices will lie on the boundary of the circle. Also, the opposite angles of square sum up to 180 degrees.

Is parallelogram a cyclic quadrilateral?

For a parallelogram to be cyclic or inscribed in a circle, the opposite angles of that parallelogram should be supplementary. Hence, not all the parallelogram is a cyclic quadrilateral.

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