Let triangle ABC and triangle DBC be equal triangles which are on the same base BC and on the same side of it. Join line AD.

I say that line AD is parallel to line BC.

If not, draw line AE through the point A parallel to the straight line BC, and join line EC.

Therefore the triangle ABC equals the triangle EBC, for it is on the same base BC with it and in the same parallels.

But triangle ABC equals triangle DBC, therefore triangle DBC also equals triangle EBC, the greater equals the less, which is impossible.

Therefore line AE is not parallel to line BC.

Similarly we can prove that neither is any other straight line except line AD, therefore line AD is parallel to line BC.

Therefore equal triangles which are on the same base and on the same side are also in the same parallels.

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