Introduction

Geometry's Long History with the Circle

Geometry is one of the oldest branches of mathematics, with roots that stretch back to ancient Egypt before flourishing further in Greece, India and China; its systematic, proof-based development is usually traced to around the 6th century BCE. The circle was one of its earliest subjects of serious study — Greek mathematicians including Thales, Menaechmus and Archimedes all worked on the circle and its tangent lines as early as the 5th century BCE, and a few decades later Euclid gathered the geometric knowledge of his time into the famous treatise The Elements. Much later, the French philosopher and mathematician René Descartes fused algebra with geometry into what we now call coordinate geometry — often called Cartesian geometry in his honour — which is the toolkit this chapter, and the whole unit that follows it, is built on.

You already meet circles constantly outside mathematics: the wheel of a bicycle or a bullock cart, a bangle, or a coin are all everyday circular shapes.

What This Chapter Covers

Using coordinate geometry, this chapter derives the equation of a circle in both the centre–radius (standard) form and the general second-degree form, and works out when a point lies inside, on, or outside a given circle. It then studies lines in relation to a circle — when a line is a tangent, and how to find the equation of a chord, tangent or normal at a point — before introducing the chord of contact, pole and polar, and finally the relative positions two circles can take with respect to each other (such as touching externally or internally) and their common tangents.