# Proportionality and Similarity

## Abstract

This chapter develops a set of tools which will allow us to start a systematic study of the *metric* aspects of Plane Euclidean Geometry; generally speaking, the central problem with which we shall be concerned here is that of comparing ratios of lengths of line segments. Among several interesting and important applications, the most prominent ones are the theorems of Thales and Pythagoras, which will reveal themselves to be almost indispensable hereafter. We also present a series of classical results, among which we highlight the study of the Apollonius circle and the solution of the Apollonius tangency problem, the collinearity and concurrence theorems of Ceva and Menelao, and some of the many theorems of Euler on the geometry of the triangle.

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