Proposition XXXIX. Theorem.
|(172)||Equal triangles (B A C and B D C) on the same base and on the same side of it are between the same parallels.|
For if the right line A D which joins the vertices of the triangles be not parallel to B C, draw through the point A a right line A E parallel to B C, cutting a side B D of the triangle B D C or the side produced in a point E different from the vertex, and draw C E.
Because the right lines A E and B C are parallel, the triangle B E C is equal to B A C (XXXVII); but B D C is also equal to B A C (hyp.), therefore B E C and B D C are equal; a part equal to the whole, which is absurd. Therefore the line A E is not parallel to B C; and in the same manner it can be demonstrated, that no other line except A D is parallel to it; therefore A D is parallel to B C.
Book I: Euclid, Book I (ed. Dionysius Lardner, 11th Edition, 1855)
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