In parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas.

Let *ACDB* be a parallelogrammic area, and *BC* its diameter.

I say that the opposite sides and angles of the parallelogram *ACDB* equal one another, and the diameter *BC* bisects it.

Since *AB* is parallel to *CD,* and the straight line *BC* falls upon them, therefore the alternate angles *ABC* and *BCD* equal one another.

Again, since *AC* is parallel to *BD,* and *BC* falls upon them, therefore the alternate angles *ACB* and *CBD* equal one another.

Therefore *ABC* and *DCB* are two triangles having the two angles *ABC* and *BCA* equal to the two angles *DCB* and *CBD* respectively, and one side equal to one side, namely that adjoining the equal angles and common to both of them, *BC.* Therefore they also have the remaining sides equal to the remaining sides respectively, and the remaining angle to the remaining angle. Therefore the side *AB* equals *CD,* and *AC* equals *BD,* and further the angle *BAC* equals the angle *CDB.*

Since the angle *ABC* equals the angle *BCD,* and the angle *CBD* equals the angle *ACB,* therefore the whole angle *ABD* equals the whole angle *ACD.*

And the angle *BAC* was also proved equal to the angle *CDB.*

Therefore in parallelogrammic areas the opposite sides and angles equal one another.

I say, next, that the diameter also bisects the areas.

Since *AB* equals *CD,* and *BC* is common, the two sides *AB* and *BC* equal the two sides *DC* and *CB* respectively, and the angle *ABC* equals the angle *BCD.* Therefore the base *AC* also equals *DB,* and the triangle *ABC* equals the triangle *DCB.*

Therefore the diameter *BC* bisects the parallelogram *ACDB.*

Therefore *in parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas.*

Q.E.D.

This proposition begins the study of areas of rectilinear figures. It is a modest beginning, but it allows the comparison of triangles and parallelograms so that problems and results concerning one can be converted to problems and results concerning the other.