(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

In any parallelogram the complements of the parallelograms about the diameter are equal to one another.

Let ABCD be a parallelogram,
and AC its diameter;

and about AC let
EH, FG
be parallelograms, and
BK, KD
the so-called complements;

I say that the complement BK
is equal to the complement KD.

For since ABCD is a parallelogram,
and AC its diameter,

the triangle ABC is equal to
the triangle ACD.
[I. 34]

Again, since EH is a parallelogram,
and AK is its diameter,

the triangle AEK is equal to
the triangle AHK.

For the same reason

the triangle KFC is also
equal to KGC.

Now, since the triangle AEK
is equal to the
triangle AHK,

and KFC to
KGC,

the triangle AEK together with
KGC is equal to the
triangle AHK together with
KFC.
[C.N. 2]

And the whole triangle ABC is also
equal to the whole
ADC;

therefore the complement BK
which remains is equal to the
complement KD which remains.
[C.N. 3]

Therefore etc. Q.E.D.

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

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