Computer algebra methods based on results of commutative algebra like Groebner bases of ideals and elimination of variables make it possible to solve complex, elementary and non elementary problems of geometry, which are difficult to solve using a classical approach. Computer algebra methods permit the proof of geometric theorems, automatic derivation and discovery of formulas, construction of objects which have given properties and which cannot be easily done with a ruler and compass, etc. On the other hand classical methods offer better insight into geometric situations, show the beauty of geometry and enable better understanding the problem. (Contains 8 figures.)