Inventing theorems with GeoGebra: a new altitude theorem

The altitude theorem states that, in a right triangle, the altitude drawn form the right angle to the hypotenuse divides the hypotenuse into two segments, the length of the altitude is the geometric mean of these two segments. A generalization of this classical theorem is discovered and proved here through the interaction of automated and human reasoning. GeoGebra automated reasoning tools reveal the existence of a large family of triangles (right and pseudo-right) that verify the thesis of the theorem. Then human reasoning leads to the characterization of pseudo-right triangles from euclidean and lorentzian geometries. Some reflections of educational nature about the development of human reasoning suported by technological tools arise naturally.