Recycler

Triangle Recycler F

This is the Moebius transformation F, which recycles the vertices of a triangle A -- > B -- > C -- > A.
Its characteristic parallelogram (by definition, having as vertices the fixed points of F and the poles of F and F^(-1)) is a rhombus, with an angle of 60 degrees. Two opposite vertices of it coincide with the isodynamic points. The two other opposite vertices are on the Lemoine axis and coincide with the inverses of the Brocard points w.r. to the circumcircle. F leaves invariant (as a whole) the bundle generated by the Apollonian circles of the triangle (recycling these circles). F leaves fixed the bundle generated from the circumcircle and the Brocard circle (bundle orthogonal to the previous). Each circle of the bundle remains invariant under F. The circumcenter O and the Brocard points build an orbit of F (O -- > G -- > F -- > O). All the "orbit-triangles" (X, F(X), F(F(X))) have the same characteristic parallelogram, hence the same Lemoine axis and Brocard diameter. The circumcircles of these triangles belong to the bundle fixed by F. The triangles having the same circumcircle and same Brocard points with ABC, coincide with the orbits of points X lying on the circumcircle. They have all the same Brocard ellipse, which coincides with the envelope of lines [Y,F(Y)], for points Y on the circumcircle. Later, because the Recycler F coincides on the circumcircle with the "Projective Recycler F* " which is defined as the Projectivity leaving invariant the circumcircle and recycling the vertices A -- > B -- > C -- > A.

The Brocard points of all the orbit-triangles lie on two circle-arcs (J'P'J) and its symmetric w.r. to JJ'. For an interesting relation of the Recycler to another subject look at file SixPivots.html .

http://users.math.uoc.gr/~pamfilos/