PROPOSITION 22. THEOREM.
If four straight lines he proportionals, the similar rectilineal figures similarly described on them shall also be proportionals; and if the similar rectilineal figures similarly described on four straight lines be proportionals, those straight lines shall be proportionals.
Let the four straight lines AB, CD, EF, GH be proportionals, namely, AB to CD as EF is to GH;
and on AB, CD let the similar rectilineal figures KAB, LCD be similarly described;
and on EF, GH let the similar rectilineal figures MF, NH be similarly described:
the figure KAB shall be to the figure LCD as the figure MF is to the figure NH.
To AB and CD take a third proportional X, and to EF and GH a third proportional O. [VI. 11.
Then, because AB is to CD as EF is to GH, [Hypothesis.
and AB is to CD as CD is to X; [Construction.
and EF is to GH as GH is to O; [Construction.
therefore CD is to X as GH is to O. [V. 11.
And AB is to CD as EF is to GH;
therefore, ex æquali, AB is to X as EF is to O. [V. 22.
But as AB is to X, so is the rectilineal figure KAB to the rectilineal figure LCD; [VI. 20, Corollary 2.
and as EF is to O, so is the rectilineal figure MF to the rectilineal figure NH; [VI. 20, Corollary 2.
