Now, ∠AOB = 180° as AB is a straight line (diameter). 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. Proof: Consider the figure below, where AB is the diameter of the circle. Theorem 2: The angle subtended by the diameter at the circumference is a right angle. 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.' Hence, using the exterior angle theorem, we get, Angles opposite to equal sides are equal) In ΔOBC, ∠OBC = ∠OCB, because OB = OC (OB and OC being the radii.In ΔOAC, ∠OAC = ∠OCA, because OA = OC (OA and OC being the radii.There are two triangles formed ΔOAC and ΔOBC. Draw a line segment through O and C, and let it intersect the circle again at point D, as shown. 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. Theorem 1: The angle subtended by a chord at the center is twice the angle subtended by it at the circumference. In this section, let us prove some of the important circle theorems discussed above. Let us now go through the statements of some important circle theorems in the next section. Arc: An arc of a circle is referred to as a curve, which is a part or portion of its circumference/boundary.Sector: A sector of a circle is the area enclosed by the two radii and arc of the circle.Segment: A segment of a circle is the area enclosed by a chord and arc of a circle.Tangent: A tangent to a circle is a line segment that touches the circle at a unique point and lies outside the circle.Circumference: The circumference of a circle is the perimeter of its boundary.Center: The center of a circle is a fixed point that is equidistant from all points on the boundary of the circle.Radius: The radius of a circle is the fixed distance between the center of the circle and any point on its boundary.The longest chord of the circle passes through the center of the circle and divided the circle into two equal parts.

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.

Let us recall the meaning of each component of a circle and some important terms related to the concept:

These theorems state important facts about different components of a circle. Circle theorems are statements in geometry that state important results related to circles.

Chord geometry in real life