(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

Equal triangles which are on equal bases and on the same side are also in the same parallels.

Let ABC, CDE be equal triangles on equal bases BC, CE and on the same side.

I say that they are also in the same parallels.

For let AD be joined;

I say that AD is
parallel to BE.

For, if not, let AF be drawn through A parallel to BE [I. 31] , and let FE be joined.

Therefore the triangle ABC is equal to
the triangle FCE;

for they are on equal bases
BC, CE
and in the same parallels
BE, AF.

But the triangle ABC is equal to
the triangle DCE;

therefore the triangle DCE
is also equal to the
triangle FCE,
[C.N. 1]

the greater to the less: which is impossible.

Therefore AF is not parallel to BE.

Similarly we can prove that neither is any other straight line
except AD;

therefore AD is
parallel to BE.

Therefore etc. Q.E.D.

Book I: Euclid, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)

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