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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.)

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  • $\begingroup$ Isn't it a fairly trivial corollary of properties of similar triangles? $\endgroup$ – IanF1 Sep 17 '15 at 13:46
  • $\begingroup$ Ignore me, i missed your equally valid point that it follows from Euclid VI.2. $\endgroup$ – IanF1 Sep 17 '15 at 13:48
  • $\begingroup$ 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. $\endgroup$ – mweiss Sep 17 '15 at 20:07
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This is essentially the "intercept theorem" in the broad formulation of older textbooks (e.g. Pogorelov's): parallels cut the opposite sides of an angle in the same ratios. That transversals form an angle is unimportant. The minimal number of parallels required to make this non-vacuous is three, making two segments, unless we use the vertex (as Wikipedia does) to make do with just two. The "tradition" attributes this theorem to Thales, but it is not listed by Eudemus, who is most reliable on early geometers, and of three accounts on which the attribution is based descriptions in two do not require it. The third is Plutarch, who, even if we take him at his word, only writes:"without trouble or the assistance of any instrument merely set up a stick at the extremity of the shadow cast by the pyramid and, having thus made two triangles by the impact of the sun's rays... showed that the pyramid has to the stick the same ratio which the shadow has to the shadow". This is the formulation in terms of similar triangles of which Elements VI.2 is a precise statement.

There is little doubt that the property in terms of cutting parallels would have been easily recognized by Greek geometers. Aristotle discusses such cutting for parallel transversals, but in relation to areas, and it occurs in vertical section when cutting pyramids and cones by planes a la Democritus. Archimedes does it in the Method, but he does not need anything like the "intercept theorem" specifically. I suspect that the earliest source for the intercept formulation will not be classical because the word "intercept" (Latin "taken in between") only came into usage c. 1400.

As for the converse, the simplest case would be the "midsegment theorem": line through the midpoints of the sides of a triangle is parallel to its base. Euclid does not state anything like this either, neither does Archimedes, but a use of it is buried in his Quadrature of the Parabola (when proving that consecutive triangles are as 1:4). Combing through proofs in Apollonius's Conic Sections one can probably find more of both since he is fond of cutting conics by multiple parallels (in anticipation of oblique coordinates), but I doubt that he so much as nods at this specifically.

Singling out these statements and giving them names most likely occured during "didactization" of Euclid by textbook authors in 18-19th centuries, if not later. What is focused on and how it is parsed was different in the age of standardized education, and this is not the only example. Playfair's axiom (two intersecting lines are not both parallel to a third) is just the contrapositive of Elements I.30, and Proclus gives a lemma that if a line cuts one of the parallels it cuts the other also. But if we want exactly the modern formulation we have to go all the way to Playfair in 1795. Similarly, Acerbi writes that "the historiographical debate about the first appearance of the focus–directrix property is biased by the relevance that this characterization has assumed in the modern characterization of conic sections. Of this property, there is no trace in Apollonius’ Conica" and it is "reduced to the status of mere functionality to a pointwise determination of a parabola" in Diocles’ On Burning Mirrors.

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