To construct a parallelogram equal to a given triangle in a given rectilinear angle.

Let *ABC* be the given triangle, and *D* the given rectilinear angle.

It is required to construct in *D* a parallelogram equal to the triangle *ABC* in the rectilinear angle.

Bisect *BC* at *E,* and join *AE.* Construct the angle *CEF* on the straight line *EC* at the point *E* on it equal to the angle *D.* Draw *AG* through *A* parallel to *EC,* and draw *CG* through *C* parallel to *EF.*

Then *FECG* is a parallelogram.

Since *BE* equals *EC,* therefore the triangle *ABE* also equals the triangle *AEC,* for they are on equal bases *BE* and *EC* and in the same parallels *BC* and *AG.* Therefore the triangle *ABC* is double the triangle *AEC.*

But the parallelogram *FECG* is also double the triangle *AEC,* for it has the same base with it and is in the same parallels with it, therefore the parallelogram *FECG* equals the triangle *ABC.*

And it has the angle *CEF* equal to the given angle *D.*

Therefore the parallelogram *FECG* has been constructed equal to the given triangle *ABC,* in the angle *CEF* which equals *D.*

Q.E.F.

In this proposition, he constructs a parallelogram that has a given angle and has the same area as a given triangle. But his goals are coming up, application of areas in I.45 and quadrature in II.14. In proposition I.45, given a rectilinear figure an equal parallelogram is constructed on a given side within a given angle. This kind of construction is called “applying” an area to a side. The area is sort of laid along the line. It may be that before Euclid the area was always applied to a rectangle along the line, but Euclid generalized the construction to parallelograms. This extra generalization is not often used.

Later, in proposition II.14 a square is constructed equal to a given rectilinear figure, a process called quadrature (making into a square) of the figure. This square is a canonical measure of the area.