Isotomic transformation in the Pseudoeuclidean plane

The pseudo-Euclidean plane PE^2 can be defined as a real projective plane P^2 where the ordered triple {; ; f, J1, J2}; ; ; is called absolute figure of the pseudo-Euclidean plane PE^2 . This absolute figure is in Cayley-Klein sense and it is consisting of a real line f, the apsolute line, and two real points J1, J2, the absolute points, lying on it. In this projective model of the pseudo-Euclidean plane the positions and properties of a point in relation to its isotomic conjugates of a triangle are discussed with use of the dynamic program The Geometer’s Sketchpad. Special attention is attached to the isotomic transformation of a line related to the triangle which showing the differences in Euclidean and pseudo-Euclidean case which occur in different positions of a triangle in relation to the absolute figure. Since that transformation is a quadratic transformation and pseudo-Euclidean plane has more curves of a second order than Euclidean plane there are some cases that can not occur in Euclidean plane. Furthermore the all curves of a second order in pseudo-Euclidean case can be sown as Euclidean circles, so the isotomic transformations of them is easier to construct in pseudo-Euclidean plane. Finally formal properties of the isotomic transformation are shown. All the results are based on the synthetic argumentation.