Cyclic Quadrilateral: Properties, Theorems and Formulas

A cyclic quadrilateral is a four-sided polygon that has all its vertices lying on the circumference of a circle.

• It is also known as an inscribed quadrilateral.
• A circle that has all the vertices of any polygon on its circumference is known as the circumscribed circle or circumcircle.
• The center of the circumcircle is called the circumcenter.

Key Terms: Quadrilateral, Cyclic Quadrilateral, Circle, Area of Quadrilateral, Diagonals, Radius, Cyclic Quadrilateral Theorems, Polygon, Circumference of a circle

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What is a Cyclic Quadrilateral?

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Let ABCD be the four vertices of a quadrilateral. If all these vertices lie on the circumference of the circle, then the quadrilateral is known as Cyclic Quadrilateral. Thus we can define it as

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Angles of Cyclic Quadrilateral

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Let ∠A, ∠B, ∠C, and ∠D be the four angles of a cyclic quadrilateral, where ∠A and ∠C are opposite, and similarly, ∠B and ∠D are opposite. Then, the sum of each pair of opposite angles in a cyclic quadrilateral is supplementary (180 degrees) i.e.

∠A + ∠C = 180^∘

And ∠B + ∠D = 180^∘

Therefore, for a cyclic quadrilateral, we have

∠A + ∠B + ∠C + ∠D = 360^∘

Hence, a cyclic quadrilateral also satisfies the quadrilateral angle sum property, which states that the sum of all angles of a quadrilateral is 360 degrees.

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Radius of Cyclic Quadrilateral

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Let a, b, c, and d are the four successive sides of a cyclic quadrilateral, and s is the semi perimeter, then the radius of the cyclic quadrilateral is given by

\(R = \frac{1}{4}\sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}\)

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Diagonals of Cyclic Quadrilaterals

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Let p and q be the diagonals of a cyclic quadrilateral, then the formula to find the diagonals is given by

\(p=\sqrt{\frac{(a c+b d)(a d+b c)}{a b+c d}} \text { and } q=\sqrt{\frac{(a c+b d)(a b+c d)}{a d+b c}}\)

Where a, b, c, and d are the sides of the cyclic quadrilateral.

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Area of Cyclic Quadrilateral

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The area of a cyclic quadrilateral is given by the formula

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

Where

• a, b, c, and d are the sides of the cyclic quadrilateral.
• s is the semiperimeter

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Properties of Cyclic Quadrilateral

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The following are the properties of a cyclic quadrilateral

• All four vertices of a cyclic quadrilateral lie on the circumference of the same circle.
• The four sides that connect the vertices and touch the circle's circumference are also the four chords of that circle.
• The sum of the two opposite angles of a cyclic quadrilateral is supplementary i.e. 180^∘.
• The four perpendicular bisectors of a cyclic quadrilateral meet at the center.
• The measure of an exterior angle of a cyclic quadrilateral at a vertex is equal to the measure of the opposite interior angle.
• The perpendicular bisectors of a cyclic quadrilateral are always concurrent.
• In a cyclic quadrilateral, the product of the two diagonals is equal to the sum of the product of opposite sides.
• On joining the midpoints of the four sides of a cyclic quadrilateral, a rectangle or a parallelogram is formed.
• The area of a cyclic quadrilateral is given by A = √(s−a)(s−b)(s−c)(s−d).
• The perimeter of a cyclic quadrilateral is 2s where s = (1/2)×(a+b+c+d) is the semi perimeter.

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Cyclic Quadrilateral Theorems

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The two important theorems on the cyclic quadrilateral are given below.

Theorem 1

The sum of either pair of opposite angles of a cyclic quadrilateral is supplementary.

Proof

Given: A cyclic quadrilateral ABCD inscribed in a circle with center O.

Construction: Join the vertices A and C with O.

Cyclic Quadrilateral Theorems

Considering the arc ABC, and using the theorem that the angle subtended by the same arc is half the angle subtended at the center, we get

∠AOC = 2∠ABC = 2α   ….(i)

Similarly, considering the arc ADC, we get

Reflex ∠AOC = 2∠ADC = 2β    ….(ii)

Using equations (i) and (ii), we get

∠AOC + Reflex ∠AOC = 360°

⇒ 2∠ABC + 2∠ADC = 360°

⇒ 2α + 2β = 360°

⇒ α + β = 180°

Similarly, if the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.

Theorem 2

In a cyclic quadrilateral, the product of the diagonals is equal to the sum of the product of its two pairs of opposite sides.

Proof

Let PQRS be a cyclic quadrilateral with PQ and RS, QR and PS being the opposite sides, and PR and QS being the diagonals respectively.

Cyclic Quadrilateral Theorems

The Product of Diagonals is,

PR x QS = (PQ x RS) + (QR x PS)

The ratio of Diagonals is given by,

PR / QS = (RS × RQ) + (PS × PQ)/ (RS × PS) + (QR × PQ)

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Solved Examples

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Things to Remember

• A cyclic quadrilateral is a type of quadrilateral with its four sides lying on the circumference of a circle.
• Most quadrilaterals, namely rectangles and squares are cyclic quadrilaterals.
• The sum of the angles of a cyclic quadrilateral is also equal to 360 degrees like all other quadrilaterals.
• A cyclic quadrilateral is also called an inscribed quadrilateral.
• The circle circumscribing a quadrilateral is called a circumcircle or circumscribed circle.
• The center of the circle circumscribing a quadrilateral is called circumcenter while the radius of the same circle is called circumradius.
• The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
• The product of diagonals of a cyclic quadrilateral is equal to the sum of the two pairs of opposite sides of the same quadrilateral.