(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

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

Let ABC, DBC
be triangles on the same base BC
and in the same parallels AD,
BC;

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

Let AD be produced in both directions
to E,
F;

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

and through C
let CF be drawn parallel to
BD.
[I. 31]

Then each of the figures EBCA,
DBCF is a parallelogram;
and they are equal,

for they are on the same base BC
and in the same parallels BC,
EF.
[I. 35]

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

And the triangle DBC is half of the parallelogram DBCF; for the diameter DC bisects it. [I. 34]

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

Therefore the triangle ABC is equal to the triangle DBC.

Therefore, etc. Q.E.D.

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

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