Volume 4: Geometry (Rigorous Elementary Mathematics)

In mathematical contests and olympiads, there are four subject areas: algebra, counting, number theory, and geometry. These areas are addressed in the four books of the Rigorous Elementary Mathematics series. While there are many books that are geared towards helping students to learn how to solve problems on contests, those books tend to assume prerequisite knowledge without proofs, and they instead focus on applying theorems to problems. This 4-volume series fills the gap in the literature by proving many of the relevant theorems in a logically sequenced framework from the ground up. Throughout this series, three ideas are reinforced: writing the same object in multiple ways, breaking up equality using antisymmetry, and using equivalence relations. It is suggested that the books be read by those who have experience with mathematical proofs and problem-solving. The ideal audience consists of teachers of mathematics who want to solidify their own fundamentals, and outstanding students who want a second look at the material. Volume 4: Geometry takes a computational approach to the subject, by utilizing trigonometry, complex numbers, and vectors. There are initial chapters on lines, angles, and congruence and similarity transformations. There are two chapters on circles, which include proofs of the inscribed angle theorem and power of a point theorem using complex numbers. Geometric objects, such as triangles, convex and concave polygons, and polyhedra are studied and classified. One of the later chapters includes a detailed study of conic sections via bivariate quadratics, including a complete proof of the classification of directrix-eccentricity-focal parameter triples for ellipses, parabolas, and hyperbolas.