BC to EF: But the triangle ABG is equal to the triangle DEF; therefore also the triangle ABC is to the triangle DEF in the duplicate ratio of BC to EF. Wherefore, Similar triangles &c. Q,E.D. COR. 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 polygons may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have; and the polygons are to one another in the duplicate ratio of their homologous sides. Let ABCDE, FGHKL be similar polygons, and let AB be the side homologous to FG: the polygons ABCDE, FGHKL may be divided into the same number of similar triangles, whereof each to each has the same ratio which the polygons have to one another; and the polygon ABCDE is to the polygon FGHKL in the duplicate ratio of AB to FG. Join BE, EC, GL, LH: Then, because the polygon ABCDE is similar to the polygon FGHKL, the angle BAE is equal to the angle GFL, and BA is to AE as GF to FL (6. Def. 1): Therefore the triangles ABE, FGL have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, and therefore they are equiangular (6. 6), and therefore (6. 4) similar to one another; therefore the angle ABE is equal to the angle FGL: But, 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 E D C M B L KH G because the triangles ABE, FGL are similar, EB is to AB as LG to FG, and also, because the polygons are similar, AB is to BC as FG to GH; therefore, ex æquali, EB is to BC as LG to GH, that is, the triangles EBC, LGH, have the sides about their equal angles proportionals, and therefore they are equiangular and similar to one another: In like manner, it may be shewn that the triangle ECD is similar to the triangle LHK: Therefore the similar polygons ABCDE, FGHKL may be divided into the same number of similar triangles. Also 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 is to the polygon FGHKL in the duplicate ratio of AB to EG. For, because the triangle ABE is similar to the triangle FGL, therefore (6. 19) ABE is to FGL in the duplicate ratio of EB to LG: For the like reason, the triangle EBC is to the triangle LGH in the duplicate ratio of EB to LG: Therefore the triangle ABE is to the triangle FGL as the triangle EBC to the triangle LGH: In like manner, the triangle EBC is to the triangle LGH as the triangle ECD to the triangle LHK: Therefore the triangle ABE is to the triangle FGL as the triangle EBC to the triangle LGH and as the triangle ECD to the triangle LHK : But (5. 12) as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents; therefore, as the triangle ABE to the triangle FGL, so is the polygon ABCDE to the polygon FGHKL: But the triangle ABE is to the triangle FGL in the duplicate ratio of AB to FG; therefore also the polygon ABCDE is to the polygon FGHKL in the duplicate ratio of AB to FG: And, in like manner, it may be proved, that similar figures of any number of sides more than three are to one another in the duplicate ratio of their homologous sides; and it has already been proved (6. 19) in the case of triangles. Wherefore, universally, Similar polygons &c. Cor. If to AB, FG, a third proportional M be taken, then (5. Def. 10) AB will be to M in the duplicate ratio of AB to FG: But any rectilineal figure upon AB will be to a similar and similarly described figure upon FG in the duplicate ratio of AB to FG: Therefore, as AB to M, so is the figure upon AB to the figure upon FG, as was before proved in the case of triangles (6. 19. Cor.): Therefore, universally, 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 rectiline l figure upon the second. PROP. XXI. 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 is similar to the figure B. Because A is similar to C, they are equiangular, and have their sides about the equal angles proportionals (6. Def. 1): Again, because B is similar to C, they are equiangular, and have their sides about the equal 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: Therefore the figures A and B are equiangular, and have their sides about the equal angles proportionals; and therefore the figure A is similar to the figure B. Wherefore, Rectilineal figures &c. PROP. XXII. Q.E.D. THEOR. If four straight lines be proportionals, and similar rectilineal figures be similarly described upon the first and second, and also upon the third and fourth, these shall also be proportionals: and if the similar rectilineal figures similarly described upon the first and second, and also upon the third and fourth, of four straight lines be proportionals, those straight lines shall also be proportionals. First, 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: the figure KAB shall be to the figure LCD as the figure MF to the figure NH. To AB, CD, take a third proportional X, and to EF, GH, a third proportional O: Then, because AB is to CD as EF to GH, and that AB is to CD as CD to X, and EF to GH as GH to O, therefore (5. 11) CD is to X as GH to O: And AB is to CD as EF to GH; therefore, ex æquali, AB is to X as EF to O: But, as AB to X, so is the figure KAB to the figure M LCD (6. 20. Cor.), and as EF to O, so is the K L A C D N E F G H P R figure MF to the figure NH; therefore the figure KAB is to the figure LCD as the figure MF to the figure NH. Next, let the figure KAB be to the figure LCD as the figure MF to the figure NH: then AB shall be to CD as EF to GH. Make (6. 12), as AB to CD, so EF to PR; and upon PR (6. 18) describe the rectilineal figure SR, similar and similarly situated to either of the figures MF, NH: Then, because AB is to CD as EF to PR, and that upon AB, CD are described the similar and similarly situated figures KAB, LCD, and upon EF, PR, the similar and similarly situated figures MF, SR, therefore KAB is to LCD as MF to SR: But (Hyp.) KAB is to LCD as MF to NH; therefore MF is to SR as MF to NH, and therefore the figures SR, NH, are equal: But they are also similar, and similarly situated; therefore PR is equal to GH: And, because AB is to CD as EF to PR, and that PR is equal to GH, therefore AB is to CD as EF to GH. Wherefore, If four straight lines &c. Q.E.D. PROP. XXIII. THEOR. Equiangular parallelograms are to one another in the ratio which is compounded of the ratios of their sides. Let AC, CF be equiangular parallelograms, having the angle BCD equal to the angle ECG: the parallelogram AC shall be to the parallelogram CF in the ratio which is compounded of the ratios of their sides. Let BC, CG be placed in a straight line; therefore DC and CE will be also in a straight line: complete the parallelogram DG, and, taking any straight line K, make (6. 12) K to L as BC to CG, and L to M as DC to CE; then the ratios of K to L and L to M are the |