Triangles which are on equal bases and in the same parallels are equal to one another.
Let ABC, DEF
be triangles on equal bases
and in the same parallels
I say that the triangle ABC is equal to the triangle DEF.
For let AD be produced in
both directions to
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
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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