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too complicated to be written here. Click on the link to download a text file. 

X(13), X(14) on K293a : X(3413) = X(1379)*, R1, S1 on K293b : X(3414) = X(1380)*, R2, S2 

X(3413) = X(1379)* is the infinite point of the Steiner ellipses focal axis and X(3414) = X(1380)* is the infinite point of the other axis. These points are also the infinite points of the Kiepert hyperbola. K293a and K293b are the orthopivotal cubics with orthopivots X(3413) and X(3414) respectively, these points are inflexion points. See the FG paper "Orthocorrespondence and orthopivotal cubics" in the Downloads page and Orthopivotal cubics in the glossary. They are both circular axial cubics with singular focus G and real asymptotes parallel to the relative axis of the Steiner ellipse. The parallel at X(110) to this axis meets the circumcircle again at R1 (or R2) and the Fermat axis at S1 (or S2) on the cubic. 
