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Euclidean geometry via isometrics
This paper studies Euclidean plane geometry from the point of view of the group of isometries of the plane. This approach allows an algebraic method of proof of theorems of plane geometry and has the advantage of avoiding the questionable technique of superposition of figures used by Euclid. Isometries are defined and classified, and it is shown that every isometry is a product of at most three line reflections. Next, by associating with each point a point reflection and with each line a line reflection it is shown how basic relations among points and lines (e.g., that a point lie on a line) can be translated into algebraic statements in the group of isometries. Using these results, algebraic proofs are given for some basic classical theorems. Also, theorems are proved concerning the concurrence of various lines associated with a triangle. Finally, isogonal conjugate points of a triangle are defined and several theorems concerning them are proved.