I'm going to answer my own question. The point of intersection between the altitude and the opposite plane of an tetrahedron should be called the center of the face. Any triangle has four centers, in center, orthocenter, circumcienter, and centrod depending on how the center is defined. With an equilateral triangle, all of the centers are in the same place, so I figure i should call it the center.
A line drawn from a tetrahedorn's apex to the opposite face, which I suppose would be called the base, should be called the height. That all seems more or less straight forward. What should one call the point of intersection between the height and the base? If there is no standard word, maybe I'll make up a name, like maybe "Eto's Point" for want of a better name.