The elements of Euclid, explained and demonstrated in a new and most easie method with the uses of each proposition in all the parts of the mathematicks / by Claude Francois Milliet D'Chales, a Jesuit ; done out of French, corrected and augmented, and illustrated with nine copper plates, and the effigies of Euclid, by Reeve Williams ...

PROPOSITION XXXIX. PROBLEM.

EQual Triangles standing on the same Base, are between the same Parallels.

If the Triangles ABC, DBC, which have the same Base BC, are equal; the Line AD drawn through the tops of those Triangles, shall be Parallel to the Base. For if AD, BC, are not Parallels; having drawn a Pa∣rallel through the Point A, the same shall fall either under AD, as AO; or above as AE. Let us suppose it falleth above, continue BD, untill it meets AE in the Point E; then draw the Line CE.

Demonstration. The Triangles ABC, EBC, are equal (by the 38^th.) seeing the Lines AE, BC, are Parallel, it is also supposed that the Triangles ABC, DBC, are equal; thence the Tri∣angles DBC, EBC, should be equal; which is impossible, the first being a part of the second. From whence I conclude that the Parallel to BC cannot be drawn above as AE.

I further say, that it cannot fall under AD, as AO; because then the Tri∣angle BOC should be equal to ABC, by consequence to the Triangle DBC, that is to say a part equal to the whole. It must then be granted that the Line AD is Parallel to BC.