Throughout this module, all geometry is assumed to be within a fixed plane. We begin by recapitulating the definition of a circle and the terminology used for circles. Spheres and cylinders are the first approximation of the shape of planets and stars, of the trunks of trees, of an exploding fireball, and of a drop of water, and of manufactured objects such as wires, pipes, ball-bearings, balloons, pies and wheels. Circles are the first approximation to the orbits of planets and of their moons, to the movement of electrons in an atom, to the motion of a vehicle around a curve in the road, and to the shapes of cyclones and galaxies.

The theoretical importance of circles is reflected in the amazing number and variety of situations in science where circles are used to model physical phenomena. Students traditionally learn a greater respect and appreciation of the methods of mathematics from their study of this imaginative geometric material.

The logic becomes more involved − division into cases is often required, and results from different parts of previous geometry modules are often brought together within the one proof. They clearly need to be proven carefully, and the cleverness of the methods of proof developed in earlier modules is clearly displayed in this module. The theorems of circle geometry are not intuitively obvious to the student, in fact most people are quite surprised by the results when they first see them. Tangents are introduced in this module, and later tangents become the basis of differentiation in calculus. Lines and circles are the most elementary figures of geometry − a line is the locus of a point moving in a constant direction, and a circle is the locus of a point moving at a constant distance from some fixed point − and all our constructions are done by drawing lines with a straight edge and circles with compasses.

Most geometry so far has involved triangles and quadrilaterals, which are formed by intervals on lines, and we turn now to the geometry of circles.

Circular geometry