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](elem.1.35)]

Moreover the triangle ABC is half of the parallelogram EBCA; for the diameter AB bisects it. [[I. 34](elem.1.34)]

And the triangle DBC is half of the parallelogram DBCF; for the diameter DC bisects it. [[I. 34](elem.1.34)]

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

Therefore the triangle ABC is equal to the triangle DBC.

Therefore etc.

Q. E. D.

Here and in the next proposition Heiberg brackets the words But the halves of equal things are equal to one another

on the ground that, since the

Common Notion which asserted this fact was interpolated at a very early date (before the time of Theon), it is probable that the words here were interpolated at the same time. Cf. note above (p. 224) on the interpolated Common Notion.