We give synthetic proofs of new results in triangle geometry, focusing especially on fixed points of certain affine maps which are defined in terms of the cevian triangles of a point P and its isotomic conjugate P′, with respect to a given triangle ABC. We give a synthetic proof of Grinberg’s formula for the cyclocevian map in terms of the isotomic and isogonal maps, and show that the complement Q of the isotomic conjugate P′ has many interesting properties. If TP is the affine map taking ABC to the cevian triangle DEF for P, it is shown that Q is the unique ordinary fixed point of TP when P does not lie on the sides of triangle ABC, its anticomplementary triangle, or the Steiner circumellipse of ABC. This paper forms the foundation for several more papers to follow, in which the conic on the 5 points A, B, C, P, Q is studied and its center is characterized as a fixed point of the map λ=TP′∘T−1Pλ=TP′∘TP−1.