Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/219

Rh therefore the parallelogram BG is equal to the parallelogram DH.

And these parallelograms are equiangular to one another; therefore the sides about the equal angles are reciprocally proportional;

therefore AB is to CD as CH is to AG.

But CH is equal to E, and AG is equal to F;

therefore AB is to CD as E is to F.

Wherefore, if our straight lines &c.

If three straight lines he proportionals, the rectangle contained by the extremes is equal to the square on the mean; and if the rectangle contained hy the extremes he equal to the square on the mean, the three straight lines are proportionals.

Let the three straight lines A, B, C be proportionals, namely, let A be to B as B is to C: the rectangle contained by A and C shall be equal to the square on B.

Take D equal to B.

Then, because A is to B as B is to C, and that B is equal to D,

therefore A is to B as D is to C.

But if four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means;

therefore the rectangle contained by A and C is equal to the rectangle contained by B and D.

But the rectangle contained by B and D is the square on B because B is equal to D;

therefore the rectangle contained by A and C is equal to the square on B.