Commentary on Proposition I.39¶
Back to Proposition I.39.
This theorem is of course the partial converse of I. 37. In I. 37 we have triangles which are (1) on the same base, (2) in the same parallels, and the theorem proves (3) that the triangles are equal. Here the hypothesis (1) and the conclusion (3) are combined as hypotheses, and the conclusion is the hypothesis (2) of I. 37, that the triangles are in the same parallels. The additional qualification in this proposition that the triangles must be on the same side of the base is necessary because it is not, as in I. 37, involved in the other hypotheses.
Proclus (p. 407, 4—17) remarks that Euclid only converts I. 37 and I. 38 relative to triangles, and omits the converses of I. 35, 36 about parallelograms as unnecessary because it is easy to see that the method would be the same, and therefore the reader may properly be left to prove them for himself.
The proof is, as Proclus points out (p. 408, 5—21), equally easy on the supposition that the assumed parallel AE meets BD or CD produced beyond D.