Department of Mathematics
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Using the concepts of point-reflections and line-reflections, we show every statement about points and lines can be formally translated into a corresponding statement about isometries of the plane. By using such translations, we convert geometric theorems about points and lines into algebraic statements about isometries and solve the resulting problems algebraically using only group theory. We will go through some famous examples of theorems in geometry and prove them algebraically. This approach was initially started by G. Thomsen and eventually developed into a full-fledged theory by F. Bachmann.

1. Irving Adler, A New Look at Geometry, New York 1966 (Dover edition 2012).
2. F. Bachmann, Aufbau der Geometrie aus dem Spiegelungswbegriff, Springer Verlag, Berlin, 1983.
3. G.Thomsen, Grundlagen der Elementargeometrie in gruppenalgebraischer Behandlung. Hamburger Math. Einzelschr. 15, (1933).