(Edited by Dionysius Lardner, 1855)

Proposition XXXIX. Theorem.

[Euclid, ed. Lardner, 1855, on `Google Books`]

(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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