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 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.

 

What is a Cyclic Quadrilateral?

In the figure given below, the quadrilateral ABCD is cyclic.

Cyclic Quadrilateral

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. 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.

Cyclic Quadrilateral Theorem Proof

Cyclic Quadrilateral Theorems

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

  • 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 ∠SAR = ∠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 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.

Cyclic Quadrilateral Examples

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 = 1800 – 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 

800 + ∠D = 180o

∠D = 1800 – 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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