Book VI. wherefore as A to B, fo B. to c. Therefore if three straight lines, &c. Q. E. D. PROP. XVIII. PROB. ON a given straight line to describe a rectilineal fi gure fimilar, and similarly situated to a given rectilineal figure. a. 23.1. 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 b. 32. 1. remaining angle AGBb. wherefore the triangle FCD is equiangular to the triangle CAB. H Н E F L B C D the remaining angle FED is equal to the remaining angle CHB, and the triangle FDE equiangular to the triangle GBH. then because the angie AGB is equal to the angle CFD, and BGH to DFE, 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 have their sides about the equal angles proportionals. because the triangles 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, so, by reason of the equiangular triangles BGH, 2.21. 5. 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 ED. Wherefore because the rectilineal • 4. 6. rectilineal figures ABHG, CDEF are equiangular, and have their Book VI. sides about the equal angles proportionals, they are similar to one UN another e. !. Def.6. Next, Let it be required to describe upon a given straight line AB, a rectilineal figure similar, and similarly situated to the rectilineal figure CDKEF of five sides. Join DE, and upon the given straight line AB describe the recti lineal 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 ABHG, 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 aequali“, GH is to HL, as FE to EK. for the same reason, AB is 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 sided figures AGHLB, CFEKD are equiangular, and have their fides about the equal angles proportionals, they are similar to one another. and in the fame manner a rectilineal figure of fix sides may be described upon a given straight line similar to one given, and so on. Which was to be done. C. 4. 6. d. 5, PROP. XIX. THEOR. SIMILA 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 side BC is homologous to EF?. the triangle ABC has to the triangle a. 12. Def. 9. DEF, the duplicate ratio of that which BC has to EF. Take BG a third proportional to BC, EFb, so that BC is to EF, b. 11. 6. as EF to BG, and join GA. then, because as AB to BC, so DE to EF; alternately, AB is to DE, as BC to EF. but as BC to EF, so c. 16.9. Book VI. is EF to BG; therefore d as AB to DE, so is EF to BG. Where: mfore the sides of the triangles ABG, DEF which are about the ed. 11. s. qual angles are reciprocally proportional. but triangles which havę the sides about two equal angles reciprocally proportional are equal D. 15. 6. to one another, therefore the triangle ABG A D portionals, the first is B G CE F 8.10.Def.5. said f to have to the third the duplicate ratio of that which it has to the second; BC there fore has to BG the duplicate ratio of that which BC has to EF. 3. 1.6. but as BC to 5G, so is 8 the triangle ABC to the triangle ABG. therefore the triangle ABC has to the triangle ABG, the duplicate ratio of that which BC has to EF. but the triangle ABG is equal to the triangle DEF; wherefore also the triangle ABC has to the triangle DEF the duplicate ratio of that which BC has to EF. therefore similar triangles, &c. Q. E, D. 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. PROP. XX THEOR. of similar triangles, having the same ratio to one another that the polygons have; and the polygons have to one another the duplicate ratio of that which their homologous fides 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 fi C. 4. 6. milar to the polygon FGHKL, the angle BAE is equal to the angle Book VI. b. 6, 6. А. M reason the tri F angle ECDE B likewise is fi L G K H Also these triangles have, each to each, the fame ratio which the Because the triangle ABE is similar to the triangle FGL, ABE has to FGL the duplicate ratio of that which the side 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,sof is the triangle BEC to the triangle f. 15. 5. GLH. Again, because the triangle EBC is similar to the triangle LGH, EBC has to LGH, the duplicate ratio of that which the side EC has to the side LH. for the same reason, 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 triangle to Book VI. triangle ECD to the triangle LHK. but it has been proved that the w triangle EBC is likewise to the triangle LGH, as the triangle ABE to the triangle FGL. Therefore as the triangle ABE is to the tri- А M F L G one of the confequents, so are all the an D С KH tecedents g. 12.5. all the consequents 8. Wherefore as the triangle ABE to the tri angle FGL, so 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 side FG. Wherefore similar polygons, &c. Q. E. D. CoR. 1. In like manner it may be proved that similar 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 universally, similar rectilineal figures are to, one another in the duplicate ratio of their homologous fides. Cor. 2. And if to AB, FG two of the homologous sides a third 1.10.Def.s. proportional M be taken, AB has h to M the duplicate ratio of that which AB has to FG. but the four fided figure or polygon upon AB has to the four sided figure or polygon upon FG likewise 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, which was also i.Cor.19.6. proved in triangles i. Therefore, universally, it is manifest, that if three straight lines be proportionals, as the first is to the third, so is any rectilineal figure upon the first, to a similar and similarly described rectilineal figure upon the second. PROP |