Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/229

Rh 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.

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.