Chapter 12: Circles

Circles covers how a circle's chords, arcs and angles relate — equal chords subtend equal angles at the centre, a perpendicular from the centre bisects a chord, and the angle at the centre is always double the angle at the circumference on the same arc — plus cyclic quadrilaterals, whose four vertices all sit on one circle, with opposite angles always summing to 180°.

The "double angle" theorem only holds when both angles are subtended by the same arc, a condition it's easy to overlook. This is one of the richest proof-based chapters in Class 9, pulling together nearly every geometry tool learned so far, and it's the direct foundation for Class 10's Tangents and Secants to a Circle.