(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

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

Let ABC, DEF
be triangles on equal bases
BC, EF
and in the same parallels
BF,
AD;

I say that the triangle ABC
is equal to the triangle DEF.

For let AD be produced in
both directions to
G,
H;

through B let
BG be drawn parallel to
CA,
[I. 31]

and through
F let FH
be drawn parallel to DE.

Then each of the figures
GBCA, DEFH
is a parallelogram;

and GBCA is equal to
DEFH;

for they are on equal bases
BC, EF
and in the same parallels
BF, GH.
[I. 36]

Moreover the triangle ABC is half of the parallelogram GBCA; for the diameter AB bisects it. [I. 34]

And the triangle ABC is half of the parallelogram DEFH; for the diameter DF bisects it. [I. 34]

[But the halves of equal things are equal to one another.]

Therefore the triangle ABC is equal to the triangle DEF.

Therefore etc. Q.E.D.

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

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