Cyclic Quadrilateral

A cyclic quadrilateral is a quadrilateral whose all four vertices lie on the circumference of a circle. The circle is called the circumscribed circle or circumcircle of the quadrilateral. The concept of a cyclic quadrilateral connects the geometry of a circle with angle properties. Understanding cyclic quadrilaterals, their properties, and related theorems is essential for solving geometry problems effectively. In this guide, you will learn about definitions, properties, and important theorems of cyclic quadrilaterals.

Table of Contents

• What is a Cyclic Quadrilateral
• Properties of Cyclic Quadrilateral
• Special Cases of Cyclic Quadrilaterals
• Solved Examples on Cyclic Quadrilateral

What is a Cyclic Quadrilateral

Definition: A cyclic quadrilateral is a four-sided polygon in which all four vertices of the polygon lie on the circumference of a single circle. A cyclic quadrilateral is a quadrilateral inscribed in a circle. All four vertices of the circle lie on the circumference of the circle. The four vertices lying on the circle are called concyclic points.

Properties of Cyclic Quadrilateral

A few of the key properties of cyclic quadrilaterals are listed below:

• In a cyclic quadrilateral, all four vertices lie on the circumference of a circle, meaning the quadrilateral is perfectly inscribed in the circle.
• The four sides of the quadrilateral act as chords of the circle, since each side connects two points on the circumference.
• The exterior angle at any vertex is equal to the interior opposite angle of the quadrilateral.
• In a cyclic quadrilateral, the diagonals follow a special relationship: the product of the diagonals is equal to the sum of the products of opposite sides.
• The perpendicular bisectors of the sides are concurrent, meaning they intersect at a single point.
• These perpendicular bisectors meet at a common point called the centre (O) of the circumcircle.
• The sum of each pair of opposite angles is 180° (supplementary). If ∠A, ∠B, ∠C, and ∠D are the angles of the quadrilateral, then: ∠A+∠C=180° and ∠B+∠D=180°

Special Cases of Cyclic Quadrilaterals

Some common quadrilaterals are always cyclic under specific conditions. Understanding these cases helps in quickly identifying cyclic quadrilaterals:

• Every rectangle and every square is a cyclic quadrilateral because their opposite angles are always equal to 90°, and hence the sum of opposite angles is always supplementary.
• An isosceles trapezium (a trapezium with equal non-parallel sides) is always cyclic, as its base angles on each parallel side are equal, and hence the opposite angles add up to 180°.
• A parallelogram is cyclic only when it is a rectangle. This is because, in a general parallelogram, opposite angles are equal but not necessarily supplementary unless each angle is 90°.
• A kite is cyclic only if one pair of opposite angles are right angles (90° each).

Solved Examples on Cyclic Quadrilateral

Example 1: In cyclic quadrilateral PQRS, ∠P = 85° and ∠Q = 75°. Find ∠R and ∠S.

Solution: Given. PQRS is a cyclic quadrilateral, ∠P = 85° and ∠Q = 75°

The sum of each pair of opposite angles is 180° for a cyclic quadrilateral.

∴ ∠P + ∠R = 180° and ∠Q + ∠S = 180°

∴ ∠R = 180° - 85° = 95° and ∠S = 180° - 75° = 105°

Example 2: In cyclic quadrilateral ABCD, ∠A = (2x + 4)° and ∠C = (3x − 14)°. Find x and both angles.

Solution: Given, ∠A = (2x + 4)° and ∠C = (3x − 14)° ∠A + ∠C = 180° (∵ sum of opposite angles of a cyclic quadrilateral is supplementary)

∴ (2x + 4)° + (3x − 14)° = 180°

(5x − 10)° = 180° x = (180+10)/5 = 38

Example 3: A quadrilateral has angles 92°, 78°, 88°, and 102°. Is it cyclic?

Solution: For a cyclic quadrilateral, the sum of each pair of opposite angles is 180°.

∴ Using trial and error, we find that 92° + 88° = 180° and 78° + 102° = 180°.

Since we found two pairs of opposite angles whose sum is 180°, the quadrilateral is cyclic.