[Axiom 1.therefore the parallelogram BG is equal to the parallelogram DH.
[VI. 14.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.
[Constr.But CH is equal to E, and AG is equal to F;
[V. 7.therefore AB is to CD as E is to F.
Wherefore, if our straight lines &c. q.e.d.
PROPOSITION 17. THEOREM.
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.
[Hyp.Then, because A is to B as B is to C,
and that B is equal to D,
[V. 7.therefore A is to B as D is to C.
[VI. 16.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.
[Construction.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.
