(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

Parallelograms which are on equal bases and in the same parallels are equal to one another.

Let ABCD, EFGH
be parallelograms which are on equal bases
BC, FG
and in the same parallels AH,
BG;

I say that the parallelogram ABCD
is equal to EFGH.

For let BE, CH be joined.

Then, since BC is equal to
FG

while FG is equal to
EH,

BC is also equal to
EH.
[C.N. 1]

But they are also parallel.

And EB, HC
join them;

but straight lines joining equal and parallel straight lines
(at the extremities which are) in the same directions
(respectively) are equal and parallel.
[I. 33]

Therefore EBCH is a parallelogram.
[I. 34]

And it is equal to ABCD;

for it has the same base BC with it, and
is in the same parallels
BC, AH
with it.
[I. 35]

For the same reason also EFGH
is equal to the same;
EBCH
[I. 35]

so that the parallelogram ABCD
is also equal to EFGH.
[C.N. 1]

Therefore etc. Q.E.D.

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

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