Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/218

194 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;

make AG equal to F, and CH equal to E;

and complete the parallelograms BG, DH.

Then, because AB is to CD as E is to F,

and that E is equal to CH, and F is equal to AG,

therefore AB is to CD as CH is to AG;

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.

But the parallelogram BG is contained by the straight lines AB and F, because AG is equal to F,

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,

and that the rectangle BG is contained by AB and F, because AG is equal to F,

and that the rectangle DH is contained by CD and E, because CH is equal to E,