- §2.1Directed SegmentsGive a segment a direction and its length gets a sign: end minus start.
- §2.2Basic Problems of the Coordinate MethodDistance, dividing points, area: three formulas turn measuring into computing.
- §2.3Choosing a Good Coordinate SystemYou place the axes. Put the origin on a key point and an axis on a key line.
- §2.4Curves and EquationsPoints on the curve satisfy it, and points satisfying it lie on the curve.
- §3.1Equations of a LineFive forms for one line; only the general form Ax+By+C=0 misses none.
- §3.2The Normal Form of a LineDrop a perpendicular from O: its direction θ and length p fix the line.
- §3.3Relations between Points and LinesDistance, side of a line, angle, parallel, perpendicular: all from A, B, C.
- §3.4Families of LinesLines through the meeting point of l₁, l₂: l₁ + λl₂ = 0; one condition fixes λ.
- §4.1The CircleComplete the square twice; the sign of D²+E²−4F gives circle, point or nothing.
- §4.2The EllipseTwo pins and a string. The sum of focal distances is 2a; b²=a²−c², e=c/a.
- §4.3The Hyperbola|PF₁−PF₂| = 2a: two branches, c²=a²+b², asymptotes along the box diagonals.
- §4.4The ParabolaEqual distances to focus and directrix give y²=2px; vary e for all three conics.
- §4.5Tangents and Normals of ConicsTangents as limits of secants, the replacement rule and the reflection property.
- §5.1Simplifying by Translating the AxesMove the origin to the center or vertex, and only squares and a constant remain.
- §5.2Simplifying by Rotating the AxesTurn the axes by θ with cot 2θ=(A−C)/B to kill xy; A+C and B²−4AC never change.
- §5.3The General Quadratic EquationB²−4AC gives the type, Δ detects degeneracy, invariants give the reduced form.
- §5.5Families of ConicsS₁+λS₂=0 passes through every common point at once; one more condition fixes λ.
- §6.1Polar CoordinatesLocate a point by “which way, how far”. Every point has many polar names.
- §6.3Equations of Curves in Polar Coordinatesρ = f(θ) is a point on a turning ray. Check symmetry, range, period; then plot.
- §6.5Simplifying Calculations and Proofs with Polar CoordinatesPole at the shared endpoint: lengths become ρ, focal chords take one line.
- §7.1Parametric Equations of CurvesA third quantity t controls x and y separately: change t and the point moves.
- §7.2Eliminating and Choosing ParametersEliminate t, then check which x and y are reached. Pick the handiest parameter.
- §7.4Simplifying Calculations and Proofs with Parametric EquationsLet t be signed distance on a line: Vieta then gives products, sums, midpoints.
- §8.1How Curves Divide the Plane into RegionsA curve cuts the plane into pieces of fixed sign; one test point decides each.
- §8.2On Extremum ProblemsSlide the level curve f = c: its last touch of the region gives the max or min.
- §9.1How Curves and Equations CorrespondGot the equation? Now check purity (no extras) and completeness (none missed).
- §9.2Methods for Locus ProblemsAsk what drives the moving point; the answer picks one of four locus methods.