Topics Covered

9 key topics in this chapter

Circles & Related Terms

Chord Properties

Perpendicular from Centre to Chord

Equal Chords & Equal Distances

Arcs & Angles at Centre

Angle Subtended by Arc

Cyclic Quadrilaterals

Opposite Angles of Cyclic Quad

Concyclic Points

Study Resources

Formula / Rule	Expression
Perp bisector	\(OM ⊥ AB ⟹ AM = MB (O = centre)\)
Equal chords	\(AB = CD ⟺ OM = ON (equal distances)\)
Central ∠ = 2 × inscribed ∠	\(∠AOB = 2·∠ACB (arc AB)\)
Angles in same segment	\(∠ACB = ∠ADB (same arc AB)\)
Semicircle angle	\(∠ACB = 90° (AB diameter)\)
Cyclic quad	\(∠A + ∠C = 180°, ∠B + ∠D = 180°\)
Exterior angle (cyclic quad)	\(∠CBE = ∠ADC (exterior = opposite interior)\)

Equal chords of a circle are equidistant from the centre. The perpendicular from the centre to a chord bisects the chord.

Angle subtended by an arc at the centre is double the angle subtended at any point on the remaining part of the circle.

Angles subtended by the same arc in the same segment are equal.

Opposite angles of a cyclic quadrilateral are supplementary (sum = 180°). The converse is also true.