Book VI. wherefore as A to B, so B to C. Therefore if three straiglat lines, &c. Q. E. D. PRO P. XVIII. PROB. figure similar, and similarly situated to a given rectilineal figure. Let AB be the given straight line, and CDEF the given rectilineal figure of four sides ; it is required upon the given straight line AB to describe a rectilineal figure similar and similarly situated to CDEF. Join DF, and at the points A, B in the straight line AB make * the angle BAG equal to the angle at C, and the angle ABG equal to the angle CDF; therefore the remaining angle CFD is equal to the remaining angle AGB h. wherefore the triangle FCD is equiangular to the triangle GAB. again, at the points G, B in the straight line GB make a the angle G E B}GII equal to the augle DFE, and the angle GBH cqual to FDE; therefore the remaining A B C D angie FED is equal to the remaining angle GHB, and the triangle FDE equiangular to the triangle GEH. then because the angle AGB is equal to the angle CFD, and BGH to DIE, the whole angle AGH is equal to the whole CFE. for the same reason, the angle ABH is equal to the angle CDE; also the angle at A is equal to the angle at C, and the angle GHB to FED. therefore the rectilineal figure ABHG is equiangular to CDEF. but likewise these figures bave their sides about the equal angles proportionals. because the țriangles GAB, FCD being equiangular, BA is to AG, as DC to CF; and because AG is to GB, as CF to FD; and as GB to GH, fo, by reason of the equiangular triangles BGH, DFE, is FD to FE; therefore, ex aequali“, AG is to GH, as CF to FE. in the same manner it may be proved that AB is to BH, as CD to DE. and GH is to HB, as FE to EDC. Wherefore because the rectilineal C. 4. 6. d. 22. 5. figures ABHG, CDEF are equiangular, and have their sides about Book VI. the equal angles proportionals, they are similar to one another Next, Let it be required to describe upon a given straight line c.1. Dcf. 4. AB, a rectilineal figure similar, and similarly situated to the rectilineal figure CDKEF of five fides. Join DE, and upon the given straight line AB describe the rectilineal figure ABHG similar and similarly situated to the quadrilateral figure CDEF, by the former case. and at the points B, H in the straight line BH, make the angle HBL equal to the angle EDK, and the angle BHL equal to the angle DEK; therefore the remaining angle at K is equal to the remaining angle at L. and because the figures ADHG, CDEF are similar, the angle GHB is equal to the angle FED, and BHL is equal to DEK; wherefore the whole angle GHL is equal to the whole angle FEK. for the same reason, the angle ABL is equal to the angle CDK. therefore the five sided figures AGHLB, CFEKD are equiangular. and because the figures AGHB, CFED are similar, GH is to HB, as FE to ED; and as HB to HL, fo is ED to EK"; therefore ex c.4.6. aequali4, GH is to HL, as FE to EK. for the fame reafon, AB is d. 22. 5, to BL, as CD to DK, and BL is to LH, as DK to KE, because the triangles BLH, DKE are equiangular. therefore because the five fided figures AGHLB, CFEKD are equiangular, and have their sides about the equal angles proportionals, they are similar to che another. and in the same manner a rectilineal figure of six fides may be described upon a given straight line similar to one given, and so on. Which was to be done, PROP. XIX. THEOR, SIM plicate ratio of their homologous fides. Let ABC, DEF be similar triangles having the angle B equal to the angle E, and let AB be to BC, as DE to EF, so that the fide BC is homologous to EF, the triangle ABC has to the triangle a.s.,Dc?.50 DEF, the duplicate ratio of that which BC has to EF, Take BG a third proportional to BC, EF ), fo that BC is to EF, b, 11. 6. as EF to BG, and join GA. then, because as AB to BC, fo DE to EF; alternately, AB is to DE, as BC to EF. but as BC to EF, ļo C. !6. so Book VI. is EF to BG; therefore d as AB to DE, so is EF to BG. wheremfore the sides of the triangles ABG, DEF which are about the ed. 11. 5. qual angles are reciprocally proportional. but triangles which have the fides about two equal angles reciprocally proportional are equal F. 15.6. to one another e. there. fore the triangle ABG A D straight lines be proporf.10.Def.s. tionals, the first is said i B G T Cor. From this it is manifest, that if three straight lines be proportionals, as the first is to the third, fo is any triangle upon the first to a similar and similarly described triangle upon the second. PRO P. XX. THEO R. SIMILAR IMILAR polygons may be divided into the same number of similar triangles, having the saine ratio to one another that the polygons have; and the polygons have to one another the duplicate ratio of that which their homologous sides have. Let ABCDE, FGHKL be similar polygons, and let AB be the homologous fide to FG. the polygons ABCDE, FGHKL may be divided into the fame number of similar triangles, whereof each to each has the same ratio which the polygons have; and the polygon ABCDE has to the polygon 'FGHKL the duplicate ratio of that which the side AB has to the side FG. Join BE, EC, GL, LH. and because the polygon ABCDE is Book VI. similar to the polygon FGHKL, the angle BAE is equal to the angle GFL ·, and BA is to AE, as GF tu FL'. wherefore because a. 1. Def.6. the triangles ABE, FGL have an angle in one equal to an angle in the other, and their fides about these equal angles proportionals, the triangle ABE is equiangular b, and therefore fimilar to the triangle b. 6.6. FGL; wherefore the angle ABE is equal to the angle FGL. and, c. 4. 6. because the polygons are similar, the whole angle ABC is equal to the whole angle FGH; therefore the remaining angle EBC is equal to the remaining angle LGH. and because the triangles ABE, FGL are fimilar, EB is to BA, as LG to GFa; and also, because the polygons are similar, AB is to BC, as FG to GH ^ ; therefore, ex aequali , EB is to BC, as LG to GH; that is, the sides about d. 22. 5. the equal angles EBC, LGH are proportionals; therefore b the triangle EBC is equiangular to the triangle LGH, and similar to ito. for the same reason the trian A M gle ECD likewise is similar? to the triangle G LIIK. therefore the fimilar polygons ABCDE, D C K. H FGHKL aredivided into the same number of similar triangles. Allo these triangles have, each to each, the same ratio which the polygons have to one another, the antecedents being ABE, EBC, ECD, and the consequents FGL, LGH, LHK. and the polygon ABCDE has to the polygon FGHKL the duplicate ratio of that which the side AB has to the homologous fide FG. Because the triangle APE is similar to the triangle FGI, ABE has to FGL the duplicate ratio of that which the fide BE has to c. 19.6. the fide GL. for the same reason, the triangle BEC has to GLH the duplicate ratio of that which BE has to GL. therefore as the triangle ABE to the triangle FGL,fof is the triangle BEC to the triangle f. 11. GLH. Again, because the triangle EBC is similar to the triangle LGH, EBC bas to LGH, the duplicate ratio of that which the side EC has to the fide LH. for the same reafon, the triangle ECD has to the triangle LHK, the duplicate ratio of that which EC has to LH, as therefore the triangle EBC to the triangle LGH, fo is f the M F Book VI. triangle ECD to the triangle LHK. but it has been proved that the triangle EBC is likewise to the triangle LGH, as the triangle A B. I G one of the confequents, so are all the antece с K H dents to all the 8. 11. 5. confequents 6. Wherefore as the triangle ABE to the triangle FGL, fo is the polygon ABCDE to the polygon FGHKL. but the triangle ABE has to the triangle FGL, the duplicate ratio of that which the side AB has to the homologous side FG. Therefore also the polygon ABCDE has to the polygon FGHKL the duplicate ratio of that which AB has to the homologous fide FG, Wherefore fimilar polygons, &c. Q. E. D. Cor. 1. In like manner it may be proved that fimilar four sided figures, or of any number of sides are one to another in the duplicate ratio of their homologous fides, and it has already been proved in triangles. Therefore univerfaliy, fimilar roetilineal fin gures are to one another in the duplicate ratio of their homologous fides. COR. 2. And if to AB, FG two of the homologous fides a h.10.Def.5. third proportional M be taken, AB hash to M the duplicate ratio of that which AB has to FG. but the four sided figure or polygon upon AB has to the four sided figure or polygon upon FG likewife the duplicate ratio of that which AB has to FG. therefore as AB is to M, so is the figure upon AB to the figure upon FG, i.Cor.19.6. which was also proved in triangles'. Therefore, universally, it is manifest, that if three straight lines be proportionals, as the first is to the third, fo is any rectilineal figure upon the first, to a similar and similarly described rectilineal figure upon the second, |