Cyclic Quadrilateral

A cyclic quadrilateral is a quadrilateral that 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. 

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 that 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 that 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 that is circumscribed in a circle is called a cyclic quadrilateral. It means that all the four vertices of the 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.

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,

∠A + ∠B + ∠C + ∠D= 360° 
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,

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:

Area of Cyclic Quadrilateral

If a,b,c and d are the sides of an inscribed quadrilateral, then its area is given by:

Where s is the semiperimeter.

s = ½(a+b+c+d)
Cyclic Quadrilateral Theorems

There are two important theorems that 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: Acyclic 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 the Cyclic quadrilateral theorem. If there’s a quadrilateral that 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

•    In a cyclic quadrilateral, the sum of a pair of opposite angles is 1800(supplementary).
•    If the sum of two opposite angles is supplementary, then it’s a cyclic quadrilateral.
•    The area of a cyclic quadrilateral is where a, b, c, and d are the four sides of the quadrilateral.
•    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 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 a 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 center O.

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°.

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. 

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 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.

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,

∠A + ∠B + ∠C + ∠D= 360° 
 

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,

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:

Area of Cyclic Quadrilateral

If a,b,c and d are the sides of a inscribed quadrialteral, then its area is given by:

Where s is the semiperimeter.

s = ½(a+b+c+d)
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

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  where a, b, c, and d are the four sides of the quadrilateral.

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 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 a 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 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.