therefore the figure KAB is to the figure LCD as the figure MF is to the figure NH. [V. 11.
Next, let the figure KAB be to the similar figure LCD as the figure MF is to the similar figure NH; AB shall be to CD as EF is to GH.
Make as AB is to CD so EF to PR : [VI. 12.
and on PR describe the rectilineal figure SR, similar and
similarly situated to either of the figures MF, NH. [VI, 18.
Then, because AB is to CD as EF is to PR,
and that on AB, CD are described the similar and simi-
larly situated rectilineal figures KAB, LCD,
and on EF, PR the similar and similarly situated recti-
lineal figures MF, SR ;
therefore, by the former part of this proposition, KAB is
to LCD as MF is to SR.
But, by hypothesis, KAB is to LCD as MF is to NH;
therefore MF is to SR as MF is to NH ; [V. 11.
therefore SR is equal to NH. [V. 9.
But the figures SR and NH are similar and similarly
situated, [Construction.
therefore PR is equal to GH.
And because AB is to CD as EF is to PR,
and that PR is equal to GH ;
therefore AB is to CD as EF is to GH. [V. 7.
Wherefore, if four straight lines &c. q.e.d.
PROPOSITION 23. THEOREM.
Parallelograms with are equiangular to one another have to one another the ratio which is compounded of the ratios of their sides.
