- Ch. 1Perimeter and Area of Circles, Sectors and SegmentsDon't grind out a shaded area. Move one piece and it often forms a whole shape.
- Ch. 2Symmetry of the CircleFold a circle on any diameter and it fits itself. That fold proves the theorem.
- Ch. 3Angles in a CircleAn inscribed angle stays put as its vertex moves; the central angle is twice it.
- Ch. 4Points, Lines and Circles: Relative PositionsPositions look like pictures, but you only compare two numbers, d and the radii.
- Ch. 5Tangents to a CircleA tangent is a secant whose two points have merged; it meets the radius at 90°.
- Ch. 6Concyclic PointsProve four points share a circle, and every equal angle on it is yours to use.
- Ch. 7The Power of a PointAny line through a fixed point cuts the circle twice; PA·PB never changes.
- Ch. 8Power of a Point: ApplicationsEach point has a power OP² − r². Equal power for two circles: the radical axis.
- Ch. 9Ptolemy's Theorem and Its ApplicationsOpposite-side products add up to ≥ the diagonal product; equal iff concyclic.
- Ch. 10Circumcenter and Incenter of a TriangleThe circumcenter serves the vertices, the incenter the sides; Euler links them.
- Ch. 11Miscellaneous Circle ProblemsMixed problems test if you spot the hidden equal angles, products or lengths.