Equal triangles which are on the same base and on the same side are also in the same parallels. | ||
Let ABC and DBC be equal triangles which are on the same base BC and on the same side of it. Join AD.
I say that AD is parallel to BC. | Post.1 | |
If not, draw AE through the point A parallel to the straight line BC, and join EC. | I.31
Post.1 | |
Therefore the triangle ABC equals the triangle EBC, for it is on the same base BC with it and in the same parallels. | I.37 | |
But ABC equals DBC, therefore DBC also equals EBC, the greater equals the less, which is impossible. | C.N.1 | |
Therefore AE is not parallel to BC.
Similarly we can prove that neither is any other straight line except AD, therefore AD is parallel to BC. | ||
Therefore equal triangles which are on the same base and on the same side are also in the same parallels. | ||
Q.E.D. |