Isotomically Conjugate Quadrilaterals

Extract

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.