We study triangles ABC and points P for which the generalized orthocenter H corresponding to P coincides with a vertex A,B, or C. The set of all such points P is a union of three ellipses minus 6 points. In addition, if TP is the affine map taking ABC to the cevian triangle DEF of P with respect to ABC, P′ is the isotomic conjugate of P, and TP′ is the affine map taking ABC to the cevian triangle of P′, then we study the locus of points P for which the map MP=Tp∘K−1∘TP′ is a translation. Here, K is the complement map for ABC, and MP is an affine map taking the circumconic of ABC for P to the inconic of ABC for P. The locus in question turns out to be an elliptic curve minus 6 points, which can be synthetically constructed using the geometry of the triangle.