If four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, then the four straight lines are proportional.

Let the four straight lines *AB, CD, E,* and *F* be proportional, so that *AB* is to *CD* as *E* is to *F.*

I say that the rectangle *AB* by *F* equals the rectangle *CD* by *E.*

Draw *AG* and *CH* from the points *A* and *C* at right angles to the straight lines *AB* and *CD,* and make *AG* equal to *F,* and *CH* equal to *E.*

Complete the parallelograms *BG* and *DH.*

Then since *AB* is to *CD* as *E* is to *F,* while *E* equals *CH,* and *F* equals *AG,* therefore *AB* is to *CD* as *CH* is to *AG.*

Therefore in the parallelograms *BG* and *DH* the sides about the equal angles are reciprocally proportional.

But those equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal, therefore the parallelogram *BG* equals the parallelogram *DH.*

And *BG* is the rectangle *AB* by *F,* for *AG* equals *F,* and *DH* is the rectangle *CD* by *E,* for *E* equals *CH,* therefore the rectangle *AB* by *F* equals the rectangle *CD* by *E.*

Next, let the rectangle *AB* by *F* be equal to the rectangle *CD* by *E.*

I say that the four straight lines are proportional, so that *AB* is to *CD* as *E* is to *F.*

With the same construction, since the rectangle *AB* by *F* equals the rectangle *CD* by *E,* and the rectangle *AB* by *F* is *BG,* for *AG* equals *F,* and the rectangle *CD* by *E* is *DH,* for *CH* equals *E,* therefore *BG* equals *DH.*

And they are equiangular. But in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional.

Therefore *AB* is to *CD* as *CH* is to *AG.*

But *CH* equals *E,* and *AG* to *F,* therefore *AB* is to *CD* as *E* is to *F.*

Therefore, *if four straight lines are proportional, then the rectangle contained by the extremes equals the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, then the four straight lines are proportional.*

Q.E.D.