Olympiad Geometry

For a point $P$ and circle, $PA\cdot PB$ is constant over lines through $P$ (negative inside). The locus of equal power to two circles is the RADICAL AXIS — a line through their intersections, found by subtracting the circle equations (for $x^2+y^2=25$ and $(x-6)^2+y^2=9$ it is $x = \tfrac{13}{3}$). Three circles' radical axes concur at the radical center.

Homothety maps a figure to a scaled copy (tangent circles, midpoints, the nine-point circle). Spiral similarity (rotation + scaling about a center) explains why two segments $AB, CD$ have a unique center of spiral similarity sending one to the other — the Miquel point of the configuration. Inversion turns circles through the center into lines: the tool for "circles tangent to circles" problems.

Coordinates, complex numbers, or barycentrics are legitimate when the configuration is computationally tame (a triangle with one circle, fixed ratios). Set up so symmetries are visible (unit circle for cyclic configurations in complex numbers), and remember a synthetic lemma often shortens the bash to two lines.