# Classical source for theorem on three parallel lines cut by two transversals

I am trying to find a classical source for the following theorem about parallel lines and transversals:

If three parallel lines are cut by two transversals, then the segments between the parallels along one of the transversals are in the same ratio as the corresponding segments along the other transversal.

(For an illustration of this theorem, see http://hotmath.com/hotmath_help/topics/parallel-lines-and-transversals.html.)

This theorem is a standard part of the conventional secondary Geometry course in the United States (and probably elsewhere?), not least because it lends itself naturally to computational problems that can easily be tested on standardized assessments. An added benefit from a pedagogical perspective is that it has a naturally-stated converse that is superficially plausible but false. (It also has a more clumsily-stated converse that is true.)

I assumed that I would find this in Euclid's Elements, but the closest I can find there is Book VI, Proposition 2, which establishes an analogous result for a line parallel to one side of a triangle and intersecting the other two sides. The theorem I am interested in can be proved fairly easily as a corollary of VI.2, but as far as I can see Euclid does not state it explicitly.

What is a classical source for this theorem? (I am also interested in any historical background on its converse.)

• Isn't it a fairly trivial corollary of properties of similar triangles? Sep 17, 2015 at 13:46
• Ignore me, i missed your equally valid point that it follows from Euclid VI.2. Sep 17, 2015 at 13:48
• Many of the propositions in Euclid are trivial consequences of basic properties. Nevertheless they are stated explicitly and proved. This one does not seem to be. Sep 17, 2015 at 20:07