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.
∠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.
∠A + ∠B + ∠C + ∠D= 360°
Cyclic Quadrilateral Theorems
Two primary theorems prove the cyclic quadrilateral.
The total sum of either pair of opposite angles is supplementary in nature, in any corresponding cyclic quadrilateral.
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.