Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/221

Rh therefore the remaining angle AGB is equal to the remain- ing angle CFD, and the triangle AGB is equiangular to the triangle CFD, Again, at the point B, in the straight line BG, make the angle GBH equal to the angle FDE; and at the point G, in the straight line BG, make the angle BGH equal to the angle DFE; [1.23. therefore the remaining angle BHG is equal to the re- maining angle DEF, and the triangle BHG is equiangular to the triangle DEF.

Then, because the angle AGB is equal to the angle CFD, and the angle BGH equal to the angle DFE; [Construction. therefore the whole angle AGH is equal to the whole angle CFE. [Axiom 2. For the same reason the angle ABH is equal to the angle CDE. And the angle BAG is equal to the angle DCF, and the angle BHG is equal to the angle DEF. Therefore the rectilineal figure ABHG is equiangular to the rectilineal figure CDEF.

Also these figures have their sides about the equal angles proportionals.

For, because the triangle BAC is equiangular to the triangle DCF, therefore BA to AC as DC is to CF. [VI. 4. And, for the same reason, AC is to GB as CF is to FD, and BC is to GH as DF is to FE ; therefore, ex aequali, AG is to GH as CF is to FB. [V. 22. In the same manner it may be shewn that AB is to BH as CD is to DE. And GH is to HB as FE is to ED. [VI. 4.

Therefore, the rectilineal figures ABHG and CDEF are equiangular to one another, and have their sides about the equal angles proportionals ; therefore they are similar to one another. [VI. Definition 1.

Next, let it be required to describe on the given straight line AB, a rectilineal figure, similar, and similarly situated, to the rectilineal figure CDKEF of five sides.