This is the third in a series of three papers that examine Euclidean geometry via complex cross-ratios. The first two papers [ibid. 52, No. 1-2, 30-54 (1996; Zbl 0860.51009); ibid., No. 3, 215-245 (1996; Zbl 0860.51010)] considered triangle shapes and triangle coordinates. In this paper the author considers the triangle coordinates of the special points of a triangle, and shows that they are functions of its shape. She then shows how these functions can be used to prove theorems about triangles.