The results of a research about Chilean teachers of Mathematics are shown in this investigation, in subject matters of Euclidean geometry, which articulates the approaches synthetically and analytically. The study presents a situation of reference that favors certain specific phases of work. The investigation sustains in the approach Mathematical Working Space and studies the personal work of the teacher with a methodological qualitative approach. The empirical evidence provides elements that allows us to characterize the paradigms in the cartesian sub-domain, which has been a contribution to the theoretical frame.