Cover -- Half-title -- Title -- Copyright -- Contents -- Figures -- Tables -- Preface -- Background -- The Elementary Geometry Trilogy -- Axioms for Writing -- The Development -- Acknowledgements -- 1 Points and Lines -- 1.1 The Vector Structure -- 1.2 Lines and Zero Sets -- 1.3 Uniqueness of Equations -- Exercise -- 1.4 Practical Techniques -- Exercises -- 1.5 Parametrized Lines -- Exercises -- 1.6 Pencils of Lines -- Exercise -- 2 The Euclidean Plane -- 2.1 The Scalar Product -- Exercise -- 2.2 Length and Distance -- Exercises -- 2.3 The Concept of Angle -- Exercise -- 2.4 Distance from a Point to a Line -- Exercises -- 3 Circles -- 3.1 Circles as Conics -- 3.2 General Circles -- Exercises -- 3.3 Uniqueness of Equations -- Exercises -- 3.4 Intersections with Lines -- Exercise -- 3.5 Pencils of Circles -- Exercises -- 4 General Conics -- 4.1 Standard Conics -- 4.2 Parametrizing Conics -- Exercises -- 4.3 Matrices and Invariants -- Exercises -- 4.4 Intersections with Lines -- Exercises -- 4.5 The Component Lemma -- Exercises -- 5 Centres of General Conics -- 5.1 The Concept of a Centre -- 5.2 Finding Centres -- Exercises -- 5.3 Geometry of Centres -- Exercise -- 5.4 Singular Points -- 6 Degenerate Conics -- 6.1 Binary Quadratics -- 6.2 Reducible Conics -- Exercises -- 6.3 Pencils of Conics -- Exercise -- 6.4 Perpendicular Bisectors -- Exercises -- 7 Axes and Asymptotes -- 7.1 Midpoint Loci -- Exercises -- 7.2 Axes -- Exercises -- 7.3 Bisectors as Axes -- Exercises -- 7.4 Asymptotic Directions -- Exercise -- 8 Focus and Directrix -- 8.1 Focal Constructions -- Exercises -- 8.2 Principles for Finding Constructions -- 8.3 Constructions for Parabolas -- Exercise -- 8.4 Geometric Generalities -- Exercise -- 8.5 Constructions of Ellipse and Hyperbola -- Exercises -- 9 Tangents and Normals -- 9.1 Tangent Lines -- Exercises -- 9.2 Examples of Tangents.

Exercises -- 9.3 Normal Lines -- Exercises -- 10 The Parabola -- 10.1 The Axis of a Parabola -- 10.2 Practical Procedures -- Exercises -- 10.3 Parametrizing Parabolas -- Exercises -- 11 The Ellipse -- 11.1 Axes and Vertices -- Exercises -- 11.2 Rational Parametrization -- Exercises -- 11.3 Focal Properties -- Exercise -- 12 The Hyperbola -- 12.1 Asymptotes -- Exercises -- 12.2 Parametrizing Hyperbolas -- Exercises -- 12.3 Focal Properties of Hyperbolas -- 13 Pole and Polar -- 13.1 The Polars of a Conic -- Exercises -- 13.2 The Joint Tangent Equation -- Exercises -- 13.3 Orthoptic Loci -- Exercises -- 14 Congruences -- 14.1 Congruences -- 14.2 Congruent Lines -- Exercises -- 14.3 Congruent Conics -- Exercises -- 14.4 The Invariance Theorem -- Exercises -- 15 Classifying Conics -- 15.1 Rotating the Axes -- Exercises -- 15.2 Listing Normal Forms -- Exercises -- 15.3 Some Consequences -- Exercises -- 15.4 Eigenvalues and Axes -- Exercises -- 16 Distinguishing Conics -- 16.1 Distinguishing Classes -- Exercises -- 16.2 Conic Sections -- 16.3 Conics within a Class -- Exercises -- 17 Uniqueness and Invariance -- 17.1 Proof of Uniqueness -- Exercises -- 17.2 Proof of Invariance -- Index.