Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/222

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Join DE, and on the given straight line AB describe, as in the former case, the rectilineal figure ABHG, similar, and similarly situated to the rectilneal figure CDEF of four sides. At the point B in the straight line BH, make the angle HBL equal to the angle EDK and at the point H, in the straight line BH make the angle BHL equal to the angle DEK; [I. 23. therefore the remaining angle at L is equal to the remaining angle at K.

Then, because the figures ABHG, CDEF are similar, the angle ABH is equal to the angle CDE; [VI. Def. 1. and the angle HBL is equal to the angle EDK [Constr. therefore the whole angle ABL is equal to the whole angle CDK, [Axiom 2. For the same reason the whole angle GHL is equal to the whole angle FEK. Therefore the five-sided figures ABHG and CDKEF are equiangular to one another.

And, because the figures ABHG and CDEF are similar, therefore AB is to BH as CD is to DE [VI. Definition 1. but BH is to BL as DE is to DK; [VI. 4. therefore, ex aequali, AB is to BL as CD is to DK. [V. 22. For the same reason, GH is to HL as FE is to EK. And BL is to LH as DK is to KE. [VI. 4.

Therefore, the five-sided figures ABLHG and CDKEF are equiangular to one another, and have their sides about the equal angles proportionals; therefore they are similar to one another. [VI. Definition 1.

In the same manner a rectilineal figure of six sides may be described on a given straight line, similar and similarly situated to a given rectilineal figure of six sides; and so on.