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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Synthetic geometry is the branch of geometry which makes use of theorems and logical arguments to draw conclusions, as opposed to analytic geometry which uses algebra to perform geometric computations and solve problems. If the axiom set is not categorical (so that there is more than one model) one has the geometry/geometries debate to settle. That''s not a serious issue for a modern axiomatic mathematician, since the implication of axiom is now a starting point for theory rather than a self-evident plank in a platform based on intuition. And since the Erlangen program of Klein the nature of any given geometry has been seen as the connection of symmetry and the content of propositions, rather than the style of development.