📚 Concepts

📚 Circle

A set of all points in a plane that are at a fixed distance (radius) from a fixed point (centre).

📚 Radius (𝑟)

The distance from the centre of the circle to any point on its boundary.

📚 Centre

The fixed point from which all points on the circle are equidistant.

📚 Chord

A line segment joining any two points on the circle.

📚 Diameter (𝑑)

A chord that passes through the centre of the circle. Diameter = 2 × 𝑟

📚 Arc

A part or segment of the circumference of a circle.

📚 Circumference

The total distance around the circle. Circumference = 2 × π × 𝑟

📚 Segment

The region between a chord and the arc.

📚 Sector

The region enclosed by two radii and the arc connecting them.

📚 Angle Subtended by a Chord at the Centre

The angle formed at the centre by lines drawn from the ends of a chord.

📚 Perpendicular from Centre to Chord

It bisects the chord (divides it into two equal parts).

📚 Equal Chords Theorem

Equal chords subtend equal angles at the centre of the circle.

📚 Concyclic Points

Points that lie on the same circle.

✍️ Theorems

✍️ Theorem 1: Equal Chords Subtend Equal Angles at the Centre

If two or more chords of a circle are equal in length, then the angles they subtend at the centre of the circle are also equal.

📌 That is, if AB = CD, then ∠AOB = ∠COD, where O is the centre of the circle.

✍️ Theorem 2: Perpendicular from Centre to a Chord Bisects the Chord

A line drawn from the centre of a circle that is perpendicular to a chord divides the chord into two equal parts.

📌 If OM ⊥ AB (a chord), then AM = MB.

✍️ Theorem 3: Converse of Theorem 2

If a line from the centre of a circle bisects a chord (not passing through the centre), then it is perpendicular to that chord.

📌 If AM = MB, and O is the centre, then OM ⊥ AB.

✍️ Theorem 4: Angles in the Same Segment are Equal

Angles formed at the circumference by the same arc (or chord) are always equal.

📌 If points A, B, and C lie on a circle, and arc AC is common, then ∠ABC = ∠ADC.

✍️ Theorem 5: Angle in a Semicircle is a Right Angle (90°)

If a triangle is formed by taking the diameter of a circle as one side and the third point on the circle, then the angle at that point is always 90°.

📌 In ∆ABC, where AC is the diameter, ∠B = 90°.

✍️ Theorem 6: Opposite Angles of a Cyclic Quadrilateral are Supplementary

If a quadrilateral is drawn such that all its vertices lie on the circle (cyclic quadrilateral), then the sum of its opposite angles is 180°.

📌 If ABCD is cyclic, then ∠A + ∠C = 180° and ∠B + ∠D = 180°.

✍️ Theorem 7: Tangent to a Circle is Perpendicular to Radius at the Point of Contact

The tangent drawn to a circle at any point is always perpendicular to the radius drawn to the point of contact.

📌 If TP is tangent at point P, and O is the centre, then OP ⊥ TP.

✍️ Theorem 8: Tangents Drawn from an External Point are Equal

The lengths of the two tangents drawn from a point outside the circle to the circle are always equal.

📌 If PA and PB are tangents from point P to the circle touching at A and B, then PA = PB.