larly situated, to the rectilineal figure CDKEF of five sides. 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; there+ 39. 1. & fore the remaining angle at K is equal to the remainS Ax. ing angle at L: and because the figures ABHG, CDEF † 1 Def.ø. 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 + 1 Def. 6. are similar, GH is to HB t, as FE to ED; but as HB to HL, so is ED to EK ;* therefore, ex æquali*, GH is to HL, as FE to EK: for the same reason, * 4. 6. 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 sides about the equal angles proportionals, they are similar to one another. In the same manner a rectilineal figure of six sides may be described upon a given straight line similar to one given, and so on. Which was to be done. * 4. 6. 29. 5. * PROP. XIX. THEOR. Similar triangles are to one another in the duplicate ratio of their homologous sides. * 12 Def. 5. 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 may be * homologous to EF: the triangle ABC shall have to the triangle DEF the duplicate ratio of that which BC has to EF. Take * BG a third proportional to BC, EF, so that BC may be to EF, 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 EFt, so is EF to BG; therefore *, as AB to DE, so is EF to BG: * 11.6. *16. 5. + Constr. * 11. 5. D 15. 6. B G therefore the sides of the triangles ABG, DEF, which А. СЕ DEF: and because as BC is to EF, so EF to BG; and that if three straight lines be proportional, the first is said to have to the third the * 10 Def.5. duplicate ratio of that which it has to the second; therefore BC has to BG the duplicate ratio of that which BC has to EF: but as BC to BG, so is * the triangle * 1. 6. 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; therefore 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, so is any triangle upon the first to a similar and similarly described triangle upon the second. PROP. XX. THEOR. similar triangles, having the same ratio to one another Let ABCDE, FGHKL be similar polygons, and let AB be the homologous side to FG: the polygons ABCDE, FGHKL may be divided into the same number of similar triangles, whereof each shall have to each the same ratio which the polygons have; and the polygon ABCDE shall have 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 similar to the polygon FGHKL, the angle BAE is equal * to the angle ĞFL, and BA is to AE*, * 1 Def. 6. 1 Def, 6. as GF to FL: therefore, because the triangles ABE, 6. 6. * 4. 6. * 4. 6. * 22. 5. A 6. 6. M F B В 4. 6. D FGL have an angle in one equal to an angle in the other, and their sides about these equal angles proportionals, the triangle ABE is equiangular* to the triangle FGL; and therefore* similar to it; wherefore the angle ABE is equal to the angle FGL: and, because the * 1 Def. 6. polygons are similar, the whole angle ABC is equal to the whole angle FGH; therefore the remaining + 3 Ax. angle EBC is equal + to the remaining angle LGH: and because the triangles ABE, FGL are similar, EB is to BA *, as LG to GF; and also because the poly1 Def. 6. gons are similar, AB is to BC*, as FG to GH; there fore, ex æquali *, EB is to BC, as LG to GH; that is, E кн Also these triangles shall have, each to each, the same ratio which the polygons have to one another, the antecedents being A BE, EBC, ECD, and the consequents FGL, LGH, LHK: and the polygon ABCDE shall have to the polygon FGHKL, the duplicate ratio of that which the side AB has to the homologous side FG. Because the triangle ABE is similar to the triangle * 19. 6. FGL, ABE has to FGL the duplicate ratio * of that which the side BE has to the side GL: for the same reason, the triangle BEC has to GLH the duplicate ratio of that which BÊ has to GL: therefore, as the triangle ABE is to the triangle FGL, so * is the triangle BEC to the triangle 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: therefore as the triangle EBC to the triangle LGH, so is * the triangle ECD to the triangle LHK: but it has been proved, that the triangle EBC is likewise to the triangle LGH, as the triangle ABE to the triangle FGL; therefore, as the triangle ABE to the triangle FGL, so * 11. 5. * 11.5. A M F * 12. 5. is triangle EBC to triangle LGH, and triangle ECD to triangle LHK: and there- E B G fore, as one of the antecedents to one of the consequents *, K H so are all the antecedents to all the consequents; that is, as the triangle ABE to the triangle FGL, so is the polygon ABČDE to the polygon FGHKL: but the triangle ABE has to the triangle FGL, the duplicate ratio of that which the † 19. 6. side A B 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 sides: and it has already been proved + in triangles : † 19. 6. therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides. Cor. 2. And if to AB, FG, two of the homologous sides, a third + proportional M be taken, AB* has to † 11. 6. . M the duplicate ratio of that which AB has to FG: *10 Def.5. but the four-sided figure or polygon upon AB, has to the four-sided figure or polygon upon FG likewise the duplicate ratiot of that which AB has to FG; there- + Cor. 1. foret, as AB is to M, so is the figure upon AB to the † 11. 5. figure upon FG: which was also proved * in triangles: *Cor.19.6. 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. XXL THEOR. Rectilineal figures which are similar to the same rectilineal figure, are also similar to one another. Let each of the rectilineal figures A, B be similar to the rectilineal figure C: the figure A shall be similar to the figure B. Because A is similar to C, they are equiangular, and also have their sides about the equal angles proportion B * 1 Def. 6. al *: again, because B is similar to C, they are equiangular, and have their sides about the equal * 1 Def.6. angles proportionals * : therefore the figures A, B, are each of them equiangular to C, and have the sides about the equal angles of each of them and of C proportionals. Wherefore the rectilineal * 1 Ax. 1. figures A and Care* equiangular, and have their sides about the equal angles * proportionals: therefore A is *1 Def. 6. similar * to B. Therefore rectilineal figures, &c. Q. E. D. # 11.5. If four straight lines be proportionals, the similar recti. lineal figures similarly described upon them shall also be proportionals: and if the similar rectilineal figures similarly described upon four straight lines be proportionals, those straight lines shall be proportionals. * 11.5. * 22. 5. 6. Let the four straight lines AB, CD, EF, GH be proportionals, viz. AB to CD, as EF to GH; and upon AB, CD let the similar rectilineal figures KAB, LCD be similarly described ; and upon EF, GH the similar rectilineal figures MF, NH, in like manner: the rec tilineal figure KAB shall be to LCD, as MF to NH. * 11. 6. To AB, CD take a third proportional* X; and to EF, GH a third proportional O: and because AB is to CD as EF to GH, therefore CD is * to X as GH to 0; wherefore, ex æquali*, as AB to X, so EF to 0: but as AB to X, so is the rectilineal figure KAB to the * 2 Cor. 20. rectilineal figure LCD, and as EF to 0, so is * the rectilineal figure MF to the rectilineal figure NH: * 11. 5. therefore, as KAB to LCD, so * is MF to NH. And if the rectilineal figure KAB be to LCD, as MF to NH; the straight line AB shall be to CD, as EF to GH. Make * as AB to CD, so EF to PR, and upon PR describe * the rectilineal figure SR similar and similarly situated to either of the figures MF, NH: then, because as AB to CD, so is EF to PR, and that upon AB, CD are described the similar and similarly situated rectilineals KAB, LCD, and upon EF, PR, in like manner, the similar rectilineals MF, SR; therefore KAB is to LCD, as MF, to SR: but by the hypothesis 12. 6. 18. 6. |