Introduction: What is a System of Circles?

So far you've studied a single circle at a time — its equation, its tangent, its normal, its pole and polar. This chapter is about two (or three) circles together, and the relationships that appear when they share the plane.

Concretely, we will look at five questions:

1. When two circles cross each other, what angle do they make at the crossing point, and when is that angle a right angle (orthogonal circles)?
2. Every point in the plane has a "power" with respect to a circle (recall $S_{11} = x_1^2+y_1^2+2gx_1+2fy_1+c$ for a point $(x_1,y_1)$ and circle $S=0$). The set of points that have equal power with respect to two circles turns out to be a straight line — the radical axis.
3. When two circles intersect at two points, the chord joining those points is the common chord — and it turns out to be exactly the radical axis.
4. Given three circles (no two of whose centres are collinear), the three radical axes — one per pair — always meet at a single point, the radical centre.
5. When a line cuts a circle at two points, the whole family of circles passing through those same two points can be written down in one formula — this lets us solve problems like "find the circle on a given chord as diameter" without ever solving for the intersection points explicitly.

Throughout, we write a circle in the general form $S \equiv x^2+y^2+2gx+2fy+c=0$, with centre $(-g,-f)$ and radius $r=\sqrt{g^2+f^2-c}$. Everything in this chapter is really coordinate geometry plus the law of cosines — no new machinery, just a new way of comparing two circles at once.