(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

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

Let ABCD, EBCF
be parallelograms on the same base
BC and in the same parallels
AF,
BC;

I say that ABCD is equal to the
parallelogram EBCF.

For, since ABCD is a
parallelogram,

AD is equal to
BC.
[I. 34]

For the same reason also

EF is
equal to BC,

so that AD is also equal to
EF;
[C.N. 1]

and DE is common;

therefore the whole AE is equal to
the whole
DF.
[C.N. 2]

But AB is also equal to
DC;
[I. 34]

therefore the two sides EA,
AB are equal to the two sides
FD, DC
respectively,

and the angle FDC is equal to
the angle EAB, the exterior
to the interior;
[I. 29]

therefore the base EB is equal to
the base FC,

and the triangle EAB will be equal to the
triangle FDC.
[I. 4]

Let DGE be subtracted
from each;

therefore the trapazium ABGD
which remains is equal to the trapezium
EGCF
which remains.
[C.N. 3]

Let the triangle GBC
be added to each;

therefore the thole parallelogram ABCD
is equal to the whole
parallelogram EBCF.
[C.N. 2]

Therefore, etc. Q.E.D.

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

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