Let the four straight lines AB, CD, E, F, be proportionals, namely, let AB be to CD as E is to F: the rectangle contained by AB and F shall be equal to the rectangle contained by CD and E.
From the points A, C, draw AG, CH at right angles to AB, CD; [I. 11.
make AG equal to F, and CH equal to E; [I. 3.
and complete the parallelograms BG, DH. [I.31 .
Then, because AB is to CD as E is to F, [Hyp.
and that E is equal to CH, and F is equal to AG, [Construction.
therefore AB is to CD as CH is to AG; [V. 7.
that is, the sides of the parallelograms BG, DH about the equal angles are reciprocally proportional;
therefore the parallelogram BG is equal to the parallelogram DH. [VI. 14.
But the parallelogram BG is contained by the straight lines AB and F, because AG is equal to F, [Construction.
and the parallelogram DH is contained by the straight lines CD and E, because CH is equal to E;
therefore the rectangle contained by AB and F is equal to the rectangle contained by CD and E.
Next, let the rectangle contained by AB and F be equal to the rectangle contained by CD and E: these four straight lines shall be proportional, namely, AB shall be to CD as E is to F.
For, let the same construction be made.
Then, because the rectangle contained by AB and F is equal to the rectangle contained by CD and E, [Hypothesis.
and that the rectangle BG is contained by AB and F, because AG is equal to F, [Construction.
and that the rectangle DH is contained by CD and E, because CH is equal to E, [Construction.
