through B draw BE parallel to CA, and through C draw
CF parallel to BD. [I. 31.
Then each of the figures EBCA, DBCF is a parallelo-
gram ; [Definition.
and EBCA is equal to DBCF, because they are on the same base BC, and between the same parallels BG, EF. [I. 35.
And the triangle ABC is, half of the parallelogram EBCA,
because the diameter AB bisects the parallelogram ; [I. 34.
and the triangle DBC is half of the parallelogram DBCF,
because the diameter DC bisects the parallelogram. [I. 34.
But the halves of equal things are equal. [Axiom 7.
Therefore the triangle ABC is equal to the triangle DBC.
Wherefore, triangles &c. q.e.d.
PROPOSITION 38. THEOREM.
Triangles on equal bases, and between the same parallels, are equal to one another.
Let the triangles ABC, DEF be on equal bases BC, EF, and between the same parallels BF, AD : the triangle ABC shall be equal to the triangle DEF,
Produce AD both
ways to the points
G,H;
through B draw BG
parallel to CA, and
through F draw FH
parallel to ED.[I.31.
Then each of the
figures GBCA, DEFH is a parallelogram. [Definition.
And they are equal to one another because they are on
equal bases BC, EF, and between the same parallels
BF, GH. [1. 36.
And the triangle ABC is half of the parallelogram GBCA,
because the diameter AB bisects the parallelogram ;[I. 34.
and the triangle DEF is half of the parallelogram DEFH,
because the diameter DF bisects the parallelogram.
But the halves of equal things are equal. [Axiom 7.
Therefore the triangle ABC is equal to the triangle DEF.
Wherefore, triangles &c. q.e.d.
