Chapter 9: Circles

A circle is a curve in which every point is equidistant from a fixed central point. It looks simple, but the relationships among its parts , chords, arcs, angles , are surprisingly rich. This chapter explores those relationships, building up to the elegant idea that the position of a point on a circle determines specific angle properties: the angle subtended by a chord at the centre, the angle subtended in the alternate segment, the right angle in a semicircle, and the equal angles in the same segment.

The central tools are the chord-distance relationships and the angle properties. You will prove that a perpendicular from the centre to a chord bisects it (and its converse), that equal chords are equidistant from the centre, and that arcs of equal length subtend equal angles at the centre.

The middle of the chapter introduces the most beautiful angle property: the angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the remaining part of the circle. This single theorem implies many corollaries , angles in the same segment are equal, the angle in a semicircle is $90^\circ$, and (as you will see) the opposite angles of a cyclic quadrilateral sum to $180^\circ$.

By the end of the chapter you should be able to look at a figure with a circle in it and read off the chord and angle relationships at a glance. The proofs are short , usually two to four lines , but the spotting is what matters.

What's inside

• Chords and the perpendicular from the centre , definition, the bisection theorem, and converse.
• Equal chords and their distances , chords equidistant from the centre are equal.
• Angle subtended by an arc , at the centre, twice the angle at the circumference.
• Angles in the same segment , equal.
• Cyclic quadrilateral , opposite angles sum to $180^\circ$.

Key results / Formula card

Memorise this card. Most circle problems collapse to one or two of these statements.