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Boor VI. of that which EC has to LH: As therefore 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 is triangle EBC to triangle LGH, and triangle ECD to triangle LHK: And therefore, as one of the antecedents to one, of the consequents, so are all the 12. 5. antecedents to all the consequents 5. Wherefore, as the triangle ABE to the triangle 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 sides, and it has already been proved in triangles. 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, 10 Def. 5. a third proportional M be taken, ABh has to M the duplicate ratio of that which AB has to FG; but the foursided 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 i Cor. 19. 6. also proved in triangles 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. 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 also have their sides about the equal angles proportionala. Again, * 1 Def. 6. because B is simi
lar to C, they are equiangular, and have their sides
about the equal
tionals. 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 B proportionals. Wherefore the rectilineal figures A and C are equiangular b, and have their ↳ 1 Ax. 1. sides about the equal angles proportionals. Therefore A 11. 5. is similar to B. Q. E. D.
PROP. XXII. THEOR.
IF four straight lines be proportionals, the similar rectilineal 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.
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 rectilineal figure KAB is to LCD, as MF to NH.
To AB, CD take a third proportionala X; and to EF, GH a third proportional O: And because AB is to CD as EF to GH, and that CD is b to X as GH to O; wherefore, ex æqualic, as AB to X, so EF to O: But as AB to X, so
BOOK VI. is the rectilineal KAB to the rectilineal LCD, and as EF to O, so is the rectilineal MF to the rectilineal NH: Therefore, as KAB to LCD, sob is MF to NH.
d 2 Cor. 20.6.
b 11. 5.
€ 12. 6.
And if the rectilineal KAB be to LCD, as MF to NH; the straight line AB is to CD, as EF to GH.
Make as AB to CD, so EF to PR, and upon PR def 18. 6. scribe the rectilineal figure SR similar and similarly situ
ated 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; KAB is to LCD, as MF, to SR; but by the hypothesis KAB is to LCD, as MF to NH; and therefore the rectilineal MF having the same ratio to each 9. 5. of the two NH, SR, these are equals to one another: They are also similar, and similarly situated; therefore GH is equal to PR: And because as AB to CD, so is EF to PR, and that PR is equal to GH; AB is to CD, as EF to GH. If therefore four straight lines, &c. Q.E.D.
PROP. XXIII. THEOR.
See N. EQUIANGULAR parallelograms have to one ano ther 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 ratio of the parallelogram AC to the parallelogram CF, is the same with the ratio which is compounded of the ratios of their sides.
b 12. 6.
d 1. 6.
• 11. 5.
Let BC, CG be placed in a straight line: therefore DC Book VI. and CE are also in a straight linea; and complete the parallelogram DG; and taking any straight line K, makeb 14. 1. as BC to CG, so K to L; and as DC to CE, so make L to M: Therefore, the ratios of K to L, and L to M, are the same with the ratios of the sides, viz. of BC to CG, and DC to CE. But the ratio of K to M is that which is said to be compounded of the ratios of K to L, and L to M: A. Def. 5. Wherefore also K has to M the ratio compounded of the ratios of the sides: And because as BC to CG, so is the parallelogram AC to the parallelogram CH; but as BC to CG, so is K to L; therefore K ise to L, as the parallelogram AC to the parallelogram CH: Again, because as DC to CE, so is the parallelogram CH to the parallelogram CF; but as DC to CE, so is L to M; wherefore L ise to M, as the parallelogram CH to the parallelogram CF: Therefore since it has been proved, that as K to L, so is the parallelogram AC to the parallelogram CH; and as L to M, so the parallelogram CH to the parallelogram CF: ex æquali, K is to M, as the pa- 22. 5. rallelogram AC to the parallelogram CF: But K has to M the ratio which is compounded of the ratios of the sides; therefore also the parallelogram AC has to the parallelogram CF the ratio which is compounded of the ratios of the sides. Wherefore equiangular parallelograms, &c. Q. E. D.
PROP. XXIV. THEOR.
THE parallelograms about the diameter of any See N. parallelogram, are similar to the whole, and to one another.
Let ABCD be a parallelogram, of which the diameter is AC; and EG, HK the parallelograms about the diameter: The parallelograms EG, HK are similar both to the whole parallelogram ABCD, and to one another.
Because DC, GF are parallels, the angle ADC is equala a 29. 1. to the angle AGF: For the same reason, because BC, EF
BOOK VI. are parallels, the angle ABC is equal to the angle AEF: and each of the angles BCD, EFG is equal to the opposite b 34. 1. angle DAB, and therefore are equal to one another: wherefore the parallelograms ABCD, AEFG, are equiangular: And because the angle ABC is equal to the angle AEF, and the angle BAC common to the two triangles BAC, 4. 6. EAF, they are equiangular to one another; therefore as AB to BC, so is AE to EF: And because the opposite sides of parallelograms are equal to one
7. 5. anotherb, ABd is to AD, as AE G to AG; and DC to CB, as GF to FE; and also CD to DA, as FG to GA: Therefore the sides of the parallelograms ABCD, AEFG about the equal angles are proportionals; and they are therefore similar to one 1 Def. 6. another: For the same reason the parallelogram ABCD is similar to the parallelogram FHCK. Wherefore each of the parallelograms GE, KH is similar to DB: But rectilineal figures which are similar to the same rectilineal £21. 6. figure are also similar to one anotherf; therefore the parallelogram GE is similar to KH. Wherefore the parallelograms, &c. Q. E.D.
PROP. XXV. PROB.
See N. To describe a rectilineal figure which shall be similar to one, and equal to another, given rectilineal figure.
2 Cor. 45. 1. a
Let ABC be the given rectilineal figure to which the figure to be described is required to be similar, and D that to which it must be equal. It is required to describe a rectilineal figure similar to ABC, and equal to D.
Upon the straight line BC describe the parallelogram BE equal to the figure ABC; also upon CE describe a the parallelogram CM equal to D, and having the angle FCE equal to the angle CBL: Therefore BC and CF are in a 29. 1. straight line b, as also LE and EM: Between BC and CF 14. 1. find a mean proportional GH, and upon GH described the rectilineal figure KGH similar and similarly situated to the figure ABC: And because BC is to GH as GH to CF; and if three straight lines be proportionals, as the first is to the