If two segments AB, XX1 of a line have their mid-points coincident, X, X1 are said to be isotomically conjugate with regard to AB, and vice versa. For brevity in this paper A, B and X, X1 will be referred to as “ isotoms ”

If pairs of isotoms XX1, YY1, ZZ1 are taken on the sides BC, CA, AB of a triangle, and if X, Y, Z are collinear, then by Menelaus' theorem X1Y1, Z1 are also collinear, and the two lines are said to be isotomically conjugate with regard to the triangle. For brevity pairs of lines related in this way will be termed “ isotomic lines”. Conversely if two transversals make XX1, YY1 isotoms with regard to two sides of a triangle then ZZ1 will be isotoms on the third side. These simple results, the source of which I have been unable to trace, are the only results on isotomically related figures made use of in the following investigation.