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The present monograph examines the capabilities of the specialized DGS Sam, set up by a team from University of Plovdiv for the needs of synthetic geometry training. In addition to the student training applications in the monograph, the potential of DGS Sam is also demonstrated for teaching pupils from 10 to 18 years of age. The possibilities of summarizing nontrivial tasks, including tasks from international mathematical Olympiads, are shown. The new "Swap finite & infinite point" function, introduced in DGS Sam and later in GeoGebra, is a powerful tool for optimizing the learning process, quickly detecting new relationships, developing creative thinking in learners . The experiments with the introduction of a heuristic element in the training show an increased interest in mathematics and a desire for self-creation. The introduction of DGS Sam at school confirmed the hypothesis that the younger generation wants to use modern technology in training and does not feel difficulties in working with infinite elements in the expanded Euclidean plane.