Circle Theorems

We study different circle theorems in geometry related to the various components of a circle such as a chord, segments, sector, diameter, tangent, etc. Before we move on to discuss the circle theorems, let us understand the meaning of a circle. A circle is a locus of points that are at a fixed distance from a fixed point on a two-dimensional plane. The fixed point is called the center of the circle and the fixed distance is called the radius.

In this article, we will explore various circle theorems that are used in geometry for solving different problems. We shall discuss the statements and proofs of the circle theorems and solve a few examples based on these theorems to understand their applications.

Circle theorems are statements in geometry that state important results related to circles. These theorems state important facts about different components of a circle. Let us recall the meaning of each component of a circle and some important terms related to the concept:

• Chord: A chord of a circle is a line segment that touches the circle at two different points on its boundary dividing the circle into two parts. The longest chord of the circle passes through the center of the circle and divided the circle into two equal parts.
• Radius: The radius of a circle is the fixed distance between the center of the circle and any point on its boundary.
• Center: The center of a circle is a fixed point that is equidistant from all points on the boundary of the circle.
• Circumference: The circumference of a circle is the perimeter of its boundary.
• Tangent: A tangent to a circle is a line segment that touches the circle at a unique point and lies outside the circle.
• Segment: A segment of a circle is the area enclosed by a chord and arc of a circle.
• Sector: A sector of a circle is the area enclosed by the two radii and arc of the circle.
• Arc: An arc of a circle is referred to as a curve, which is a part or portion of its circumference/boundary.

Let us now go through the statements of some important circle theorems in the next section.

Now that we have understood the meaning of important terms related to a circle, let us go through the circle theorems statements below:

• The angle subtended by a chord at the center is twice the angle subtended by it at the circumference.
• The angle subtended by the diameter at the circumference is a right angle.
• The angles subtended at the circumference by the same arc are equal.
• Two equal chords subtend equal angles at the center of the circle.
• If the angles subtended by two chords at the center are equal, then the two chords are equal.
• The opposite angles in a cyclic quadrilateral are supplementary.
• The angle between the radius and the tangent at the point of contact is 90 degrees.

In this section, let us prove some of the important circle theorems discussed above.

Theorem 1: The angle subtended by a chord at the center is twice the angle subtended by it at the circumference.

Proof: Consider the following circle, in which an arc (or segment) AB subtends ∠AOB at the center O and ∠ACB at a point C on the circumference. We have to prove that ∠AOB = 2 × ∠ACB. Draw a line segment through O and C, and let it intersect the circle again at point D, as shown.

There are two triangles formed ΔOAC and ΔOBC. So, we make the following observations.

• In ΔOAC, ∠OAC = ∠OCA, because OA = OC (OA and OC being the radii. Angles opposite to equal sides are equal)
• In ΔOBC, ∠OBC = ∠OCB, because OB = OC (OB and OC being the radii. Angles opposite to equal sides are equal)

Hence, using the exterior angle theorem, we get,

∠AOD= 2×∠ACO ⋯ (1)

∠DOB= 2×∠OCB ⋯ (2)

Add equations (1) and (2):

∠AOD+ ∠DOB= 2× (∠ACO+ ∠OCB)

⇒ ∠AOB= 2× ∠ACB

Hence, we have proved the circle theorem 'The angle subtended by a chord at the center is twice the angle subtended by it at the circumference.'

Theorem 2: The angle subtended by the diameter at the circumference is a right angle.

Proof: Consider the figure below, where AB is the diameter of the circle. We need to prove that ∠ACB= 90°

Using theorem 1 'The angle subtended by a chord at the center is twice the angle subtended by it at the circumference.', we have ∠AOB= 2× ∠ACB. Now, ∠AOB = 180° as AB is a straight line (diameter). So, 2 × ∠ACB = 180° which implies ∠ACB = 90°. Hence we have proved the circle theorem 'The angle subtended by the diameter at the circumference is a right angle'.

Theorem 3: The angles subtended at the circumference by the same arc are equal.

Proof: Consider the following figure, which shows an arc AB subtending angles ACB and ADB at two arbitrary points C and D on the circumference. O is the center of the circle.

We need to prove that ∠ACB= ∠ADB.

Using the circle theorem 'The angle subtended by a chord at the center is twice the angle subtended by it at the circumference.', we have that

∠ACB= 1/2× ∠AOB ⋯ (1)

∠ADB= 1/2× ∠AOB ⋯ (2)

From equations (1) and (2), we get ∠ACB= ∠ADB. Since angles ACB and ADB are arbitrary angles, therefore, the result is true for all angles subtended by the same arc. Hence we have proved the circle theorem 'The angles subtended at the circumference by the same arc are equal.'

Theorem 4: Two equal chords subtend equal angles at the center of the circle.

Proof: Consider a circle given below with center O and two chords AB and CD such that AB = CD. Now, we need to prove ∠AOB = ∠COD.

In triangles AOB and COD, we have

OA = OC (Radii)

OB = OD (Radii)

AB = CD (Given)

So, triangles AOB and COD are congruent by SSS congruence rule. So, we have ∠AOB = ∠COD (Corresponding parts of congruent triangles).

Theorem 5: If the angles subtended by two chords at the center are equal, then the two chords are equal.

Proof: Consider a circle given below with center O and two chords AB and CD such that ∠AOB = ∠COD. Now, we need to prove AB = CD.

In triangles AOB and COD, we have

OA = OC (Radii)

OB = OD (Radii)

∠AOB = ∠COD (Given)

So, triangles AOB and COD are congruent by SAS congruence rule. So, we have AB = CD (Corresponding parts of congruent triangles).

Important Notes on Circle Theorems

• Circle theorems are statements in geometry that state important results related to circles that are used to solve various questions in geometry.
• Circle theorems in geometry are related to the various components of a circle such as a chord, segments, sector, diameter, tangent, etc.
• Angles in the same segment of a circle are equal.

☛ Related Articles:

• Vertical Angles
• Supplementary Angles Formula
• Circle Formulas

What are Circle Theorems in Geometry?

Circle theorems are statements in geometry that state important results related to circles. These theorems state important facts about different components of a circle such as a chord, segments, sector, diameter, tangent, etc.

What Does Subtend Mean in Circle Theorems?

Subtend in circle theorems means an angle that is subtended opposite an arc or a chord. When we say that an angle is subtended by an arc, this implies that the angle is made by the lines from the arc on the opposite side of the circle.

Why Do We Need Circle Theorems?

We need circle theorems to solve various problems in geometry. When we draw angles and lines inside a circle, we can deduce various patterns and theorems from it which are helpful in practical and theoretical life.

How to Find Angles in Circle Theorems?

We can find angles in circles using the circle theorems based on angles. Some of the important circle theorems based on angles are:

• The angle subtended by a chord at the center is twice the angle subtended by it at the circumference.
• The angle subtended by the diameter at the circumference is a right angle.
• The angles subtended at the circumference by the same arc are equal.

List Circle Theorems Statements.

The circle theorems are statements that state results about various components of circle. Some of the important circle theorems statements are:

• The angle subtended by a chord at the center is twice the angle subtended by it at the circumference.
• The angle subtended by the diameter at the circumference is a right angle.
• The angles subtended at the circumference by the same arc are equal.
• Two equal chords subtend equal angles at the center of the circle.
• If the angles subtended by two chords at the center are equal, then the two chords are equal.
• The opposite angles in a cyclic quadrilateral are supplementary.
• The angle between the radius and the tangent at the point of contact is 90 degrees.

How to Prove Circle Theorems?

We can prove circle theorems by using various results in geometry such as the triangle sum theorem, other circle theorems, and theorems based on angles in geometry.

Is Circle Theorems Part of Geometry?

Yes, circle theorems are part of geometry. Since a circle is a two-dimensional plane figure and plays an important role in geometry. So, all circle theorems are an important part of geometry.