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This textbook presents various techniques of elimination based on Grobner bases to prove well-known geometrical theorems and formulas. Besides proving theorems, these methods are used to discover new formulas, solve geometric inequalities, and construct objects - which cannot easily be done with a ruler and compass. Each problem is firstly solved by the method of automatic theorem proving. Secondly, problems are solved classically - without using computer where possible - so that readers can compare the strengths and weaknesses of both approaches.