What are Quadrilaterals? Explain Cyclic Quadrilaterals

A quadrilateral with all four vertices on a circle is called a cyclic quadrilateral. An inscribed quadrilateral is another name for it. The circumcircle or circumscribed circle is a circle that contains all of the vertices of any polygon on its circumference.

What is a Quadrilateral?

A quadrilateral is a four-sided polygon with four finite line segments on each respective side. The term quadrilateral is made up of two Latin words: Quadri, which means “four,” and latus, which means “hand.” It is a two-dimensional figure with four vertices and four sides (or edges). The position of all given points in a plane that are equidistant from a fixed point is named a circle.

ABCD is a cyclic quadrilateral if all four vertices of the quadrilateral are on the circumference of the circle. In other words, the vertices of a cyclic quadrilateral are formed by joining any four points on the circumference of a circle. It can be seen as a quadrilateral inscribed in a circle, with all four vertices laying on the circumference of the circle.

What is a Cyclic Quadrilateral?

A cyclic quadrilateral is a quadrilateral that is circumscribed in a circle, as per the definition. It implies that the quadrilateral’s four vertices are all within the circle’s circumference. Let’s look at a diagram and see what we’re talking about.

Cyclic Quadrilateral Angles

The total sum of the present opposite angles in a cyclic quadrilateral is supplementary in nature. 

Let us consider ∠A, ∠B, ∠C, and ∠D as the four angles of an inscribed quadrilateral. 

Then,

∠A + ∠C = 180° 

∠B + ∠D = 180°

This results in an inscribed quadrilateral also meet the property of the angle sum property of a quadrilateral. According to this property, the total sum of all the angles equals 360 degrees.

Hence,

∠A + ∠B + ∠C + ∠D= 360° 

Cyclic Quadrilateral Theorems

Two primary theorems prove the cyclic quadrilateral.

Theorem 1

The total sum of either pair of opposite angles is supplementary in nature, in any corresponding cyclic quadrilateral.

Theorem 2

The Cyclic quadrilateral theorem can be used to calculate the ratio between the diagonals and the sides. The acquired product of the diagonals of a quadrilateral etched in a circle is equal to the sum of the obtained product of its two pairs of opposite sides.

Properties of Cyclic Quadrilateral

• In any respective cyclic quadrilateral, the total sum of a pair of opposite angles is 180 that means supplementary in nature.
• If the total sum of two opposite angles in any quadrilateral is supplementary in nature, then it can be named a cyclic quadrilateral.
• The area of a cyclic quadrilateral can be founded by: Area = √(s−a)(s−b)(s−c)(s−d), where a, b, c, and d are taken as the four sides of the quadrilateral.

• The four vertices of a cyclic quadrilateral lay in the circumference of the circular shape.
• To prove a rectangle or a parallelogram, one can join the midpoints of the four sides in sequence.
• If PQRS is examined as a cyclic quadrilateral, then we can say that ∠SPR = ∠SQR, ∠QPR = ∠QSR, ∠PQS = ∠PRS, ∠QRP = ∠QSP.
• If T is taken as a point of intersection of the two diagonals in a shape, PT X TR = QT X TS.
• The exterior angle obtained if any one side of the cyclic quadrilateral produced is equal to the interior angle opposite to the other.
• In a given cyclic quadrilateral, d1 / d2 = total sum of the product of opposite sides, that share the diagonal endpoints in any quadrilateral.
• If it is taken as a cyclic quadrilateral, then the perpendicular bisectors will be concurrent without fail.
• In any cyclic quadrilateral, the four perpendicular bisectors of the given four sides in a quadrilateral meet at O that symbolizes the center of the shape.

These were a few primary and essential characteristics of a cyclic quadrilateral.