Cyclic Properties of Circle: Theorem, Properties & Examples

The set of all points in the plane that are a fixed distance (the radius) from a fixed point (the centre) is called a circle. A quadrilateral inscribed in a circle is known as a cyclic quadrilateral. That is, a circle passes through each of the quadrilateral’s four vertices. Concyclic vertices are those that are arranged in a circular pattern. Let us discuss more on the Cyclic Properties of Circles in this article.

What is a Circle?

A circle is a two-dimensional figure made by a set of points in the plane that are distanced by a constant or fixed length (radius). The origin or centre of the circle is the fixed point, and the radius is the fixed distance between the points from the origin. Every point on the circle is equidistant from a single point known as the circle’s centre.

Properties of Circle

1. If two circles have the same radius, they are said to be congruent.
2. Equidistant chords are always equidistant from the circle’s centre.
3. A chord’s perpendicular bisector crosses through the circle’s centre.
4. The line connecting the intersecting points of two circles will be perpendicular to the line connecting their centre points.
5. The radius is perpendicular to the tangent.
6. Two tangents can be drawn from an external point on a circle.

Learn Properties of Circles

Cyclic Properties of a Circle

1. When the vertices of a quadrilateral lie on the circumference of a circle, it is called a cyclic quadrilateral.
2. Concylic points are points that are located on the same circle’s circumference.

Cyclic Quadrilateral

“Cyclic” is derived from the Greek word “kuklos,” meaning “circle” or “wheel.” The term “quadrilateral” comes from the Latin word “Quadri,” which means “four sides”.

A quadrilateral inscribed in a circle is known as a cyclic quadrilateral. That is, a circle passes through each of the quadrilateral’s four vertices. Concyclic vertices are those that are arranged in a circular pattern.

Angle and Cyclic Properties of Circle

Theorems based on cyclic properties are:

Theorem -1: The opposite angles of a cyclic quadrilateral (quadrilateral inscribed in a circle) are supplementary. 

\(A B C D\) is a cyclic quadrilateral.

Therefore,

\(\angle A+\angle C=180^{\circ}\)

\(\angle B+\angle D=180^{\circ}\)

Theorem-2: The exterior angle of a cyclic quadrilateral is equal to the opposite angle of its supplementary interior angle.

\(A B C D\) is a cyclic quadrilateral.

So, \(\angle a=\angle e\)

Properties of Cyclic Quadrilateral

1. In a cyclic quadrilateral, the opposite angles are supplementary.
2. The sum of the opposite angles of a cyclic quadrilateral is supplementary, \(\mathrm{i}, \mathrm{e}\), it is equal to \(180^{\circ}\).
   Let \(\angle A, \angle B, \angle C\), and \(\angle D\) be the four angles of an inscribed quadrilateral. Then,
   \(\angle A+\angle C=180^{\circ}\)
   \(\angle B+\angle D=180^{\circ}\)
3. As a result, an inscribed quadrilateral also satisfies the quadrilateral’s angle sum property, which states that the total of all angles equals \(360\) degrees.
   Hence, \(\angle A+\angle B+\angle C+\angle D=360^{\circ}\)
4. If one side of a cyclic quadrilateral is produced, the exterior angle will equal the opposite interior angle. So, an exterior angle is equal to the interior opposite angle.
   Therefore, \(\angle \beta=\angle \alpha\)
5. If the sum of any pair of opposite angles of a quadrilateral is \(180^{\circ}\), then the quadrilateral is cyclic.

Properties of Cyclic Quadrilateral Inscribed in a Circle

Any four-sided figure with all its vertices on a circle is called an inscribed quadrilateral. (The sides are thus the chords in the circle). This conjecture establishes relationships between the quadrilateral’s opposite angles. It says that these opposite angles are in supplements for each other.

The angle sum property of a quadrilateral states that the sum of all the four angles in any quadrilateral is \(360^{\circ}\). Now, if the quadrilateral is inscribed by a circle, we note an interesting property in the sum of their opposite angles. We observe that the sum of the opposite angles is \(180^{\circ}\). This means,
\(\angle A+\angle C=\angle B+\angle D=180^{\circ}\)

The Diagonal Properties of a Cyclic Quadrilateral

Ratio of Diagonals of a Cyclic Quadrilateral

In a cyclic quadrilateral \(ABCD\), the ratio of the diagonals equals the ratio of the sum of products of the sides that share the diagonals endpoints.

Therefore,

\(\frac{A C}{B D}=\frac{(A B \times A D+C B \times C D)}{(A B \times B C+A D \times D C)}\)

\(\Rightarrow \frac{\text { diagonal } 1}{\text { diagonal } 2}=\frac{(a \times d+b \times c)}{(a \times b+d \times c)}\)

\(\Rightarrow \frac{d_{1}}{d_{2}}=\frac{(a d+b c)}{(a b+d c)}\)

Product of Diagonals

If a quadrilateral is inscribed inside a cycle, the sum of the two pairs of opposite sides of the cyclic quadrilateral is equal to the product of the diagonals of the cyclic quadrilateral.

From the above cyclic quadrilateral \(ABCD\),

\((A C \times B D)=(A B \times A D)+(B C \times C D)\)

\(\Rightarrow d_{1} \times d_{2}=(a \times d)+(b \times c)\)

\(\Rightarrow d_{1} d_{2}=a d+b c\)

Learn About Different Parts of a Circle

Solved Examples – Cyclic Properties of Circle

Q.1. \(A B C D\) is a cyclic quadrilateral in which \(A C\) and \(B D\) are its diagonals. If \(\angle D B C=50^{\circ}\) and \(\angle B A C=45^{\circ}\), find \(\angle B C D=?\)

Ans: \(\angle C A D=\angle D B C=50^{\circ}\) (Angles in the same segment are equal)
Therefore, \(\angle D A B=\angle C A D+\angle B A C=50^{\circ}+45^{\circ}=95^{\circ}\)
But \(\angle D A B+\angle B C D=180^{\circ}\) (Opposite angles of a cyclic quadrilateral)
So, \(\angle B C D=180^{\circ}-95=85^{\circ}\)
Therefore, \(\angle B C D=85^{\circ}\)

Q.2. In the figure given below, find the value of \(a\) and \(b\).

Ans: By the exterior angle property of a cyclic quadrilateral, if one side of a cyclic quadrilateral is produced, the exterior angle equals the interior opposite angle.
Therefore, we get \(b=100^{\circ}\)
And \(a+30^{\circ}=60^{\circ}\)
\(\Rightarrow a=60^{\circ}-30^{\circ}=30^{\circ}\)
Therefore, the values of \(a\) and \(b\) are \(30^{\circ}\) and \(100^{\circ}\), respectively.

Q.3. Find the value of angle \(\angle S\) of a cyclic quadrilateral \(P Q R S\) if angle \(\angle Q\) is \(70^{\circ}\).
Ans: If \(PQRS\) is a cyclic quadrilateral, the sum of a pair of two opposite angles will be \(180^{\circ}\).
\(\Rightarrow \angle Q+\angle S=180^{\circ}\)
\(\Rightarrow 70^{\circ}+\angle S=180^{\circ}\)
\(\Rightarrow \angle S=180^{\circ}-70^{\circ}\)
\(\Rightarrow \angle S=110^{\circ}\)
The value of angle \(\angle S\) is \(110^{\circ}\).

Q.4. Find the value of angle \(D\) of a cyclic quadrilateral \(A B C D\) if angle \(B\) is \(65^{\circ}\).
Ans: If \(A B C D\) is a cyclic quadrilateral, the sum of a pair of two opposite angles will be \(180^{\circ}\).
\(\Rightarrow \angle B+\angle D=180^{\circ}\)
\(\Rightarrow 65^{\circ}+\angle D=180^{\circ}\)
\(\Rightarrow \angle D=180^{\circ}-65^{\circ}\)
\(\Rightarrow \angle D=115^{\circ}\)
The value of angle \(D\) is \(115^{\circ}\).

Q.5. In the figure given below, \(\angle a=60^{\circ}\), then find the \(\angle e\).

Ans: From the given \(\angle a=60^{\circ}, \angle e=\) ?
We know, by the exterior angle property of a cyclic quadrilateral, if one side of a cyclic quadrilateral is produced, then the exterior angle equals the interior opposite angle.
So, \(\angle a=\angle e\)
\(\Rightarrow \angle a=\angle e=60^{\circ}\)
Therefore, \(\angle e=60^{\circ}\).

Summary

A cyclic quadrilateral is a quadrilateral that is encompassed by a circle. A circle goes through each of the quadrilateral’s four vertices, in other words. In a cyclic quadrilateral, the total of each pair of opposite angles is \(180^{\circ}\).

If a quadrilateral has one set of opposite angles that add up to \(180^{\circ}\), it is cyclic. This article includes the definition of a circle, properties of a circle, cyclic quadrilateral, properties of a cyclic quadrilateral, angles of a cyclic quadrilateral, properties of a cyclic quadrilateral inscribed in a circle.

This article, “Cyclic Properties of Circle”, helps in understanding these concepts in detail, and it also helps us solve the problems based on these topics very easily.

Frequently Asked Questions (FAQs)

The answers to the most frequently asked questions on cyclical properties of circles are provided here:

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