This is a thorough analysis of the fundamentals of plane geometry - Euclidean, spherical and projective. The reader is provided with an abundance of geometrical facts such as classical results of plane Euclidean and non-Euclidean geometry, investigation of conic sections, projective spaces, Galilean transformations, Lorentz transformations, and Minkowskii geometry to name a few. The author considers a range of interesting problems which one meets in real life - the number of images one sees in a pair of mirrors, how to detect the position of an enemy ship from two different listening posts and the distance between two cities knowing their latitude and longitude. This book provides fresh examples, interesting exercises, many illustrations, and is pleasant to read.