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Book VI. is EF to BG; therefore d as AB to DE, so is EF to BG. where
fore the sides of the triangles ABG, DEF which are about the em 4. !1. So
qual angles are reciprocally proportional. but triangles which have
the fides about two equal angles reciprocally proportional are equal 6.
to one another e, there.
straight lines be proporfr 19. Def. 5. tignals, the first is said
E. F to have to the third the duplicate ratio of that which it has to the second; BC therefore has to BG the duplicate ratio of that which BC has to EF. but as BC to BG, so is & the triangle ABC to the triangle ABG. there, fore 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 fimilar triangles, &c. Q. E. D.
Cor. From this it is manifeft, that if three straight lines be proportionals, as the first is to the third, so is any triangle upon the fierft to a similar and similarly described triangle upon the seconde
IMILAR polygons may be divided into the same
number of similar triangles, having the same ra, dio 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 bę the homologous side 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 ; and the poly, gon ABCDE has to the polygon FGHKL the duplicate ratio of that which the side AB has to the fide FG,
Join BE, EC, GL, LH. and because the polygon ABCDE is Brok VI. fimilar to the polygon FGHKL, the angle BAE is equal to the angle GFL, and BA is to AE, as GF to FL . wherefore because a. 1. Def. 6. the triangles ABE, FGL have one angle in one equal to an angle in the other, and their fides about these equal angles proportionals, the triangle ABE is equiangular, 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 similar, EB is to BA, as LG to GF#; 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. So the equal angles EBC, LGH are proportionals; therefore the triangle EBC is equiangular to the triangle LGH, and similar to it c. for the fame reason the trian,
А gle ECD like,
M wife is similar to the triangle E LHK. therefore the similar poly. gons ABCDE, FGHKL are dir
KH vided into the same number of similar triangles.
Also these triangles have, each to each, the fame ratio which the polygons have to ane apother, the antecedents being ABE, EBC, ECD, and the consequents FGL, LGH, LHK. and the polygon ABCDE has to the palygon FGHKL thę duplicate ratio of that which the side AB has to the homologous fide FG.
Because the triangle ABE is similar to the triangle FGL, ABE has to FGL the duplicate ratio of that which the fide BE has to es the side 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 ABEto the triangleFGL, sofis the triangle BECto the triangle f. 11. Se GLH. Again, because the triangle EBC is similar to the triangle LGH, EBC has to LGH, the duplicate ratio of that which the fide 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 H. as therefore the triangle ELÇ to the triangle LGH, so iş f the
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
M as one of the
E antecedents to
G one of the confequents, fo are all the anteceD
K H dents to all the 8. 12. 5. 5
consequents &. 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 fide AB has to the homologous fide 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 similar polygons, &c. Q. E. D.
Cor. 1. In like manner it may be proved that fimilar four sided figures, or of any number of fides 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 sides.
Cor. 2. And if to AB, FG two of the homologous fides a h.co.Def.5. third proportional M be taken, AB has b 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 likewise the duplicate ratio of that which AB has to FG. therefore
as AB is to M, fo is the figure upon AB to the figure upon FG, i.Cor.19.6. which was also proved in triangles i. 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 fimilarly described rectilineal figure upon the second.
ECTILINEAL 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 fides about the equal angles proportionals 2. Again, because a. 1. Def.6. B is similar to C, they are equiangular, and have their fides about the equal angles proportion
B als a. 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 figures A and B are equiangular b, and have their fides about b. s. As. I. the equal angles proportionals c. Therefore A is similar a to B. C. 11. 5. R. E. D.
(F four straight lines be proportionals, the fimilar rec
tilineal figures similarly described upon them shall also be proportionals. and if the similar rectilineal figures fimilarly 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 proportional a X; and to EF, GH à 2. 11. 6. third proportional O. and because AB is to CD, as EF to GH, b. 11. 5. therefore CD is o to X, as GH to 0; wherefore ex aequali“, as AB c. 22. 5
b. II. 5.
Book VI. to X, so EP to 0. but as AB to X, so is d the rectilineal KAB to
the rectilineal LCD, and as EF to O, so is the rectilineal MF to d. 2. Cor.
the rectilineal NH. therefore as KAB to LCD, so bis MF to NH.
And if the rectilineal KAB be to LCD, as MF tỌ NH; the
straight line AB is to CD, as EF to GH. e. 12. 6.
Make e as AB to CD, fo EF to PR, and upon PR describe! f. 1$. 6. the rectilineal figure SR similar and similarly situated to either of
PROP. XXIII. THEOR,
have to one another the ratio which is compounded of the Tatios of their fides
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 com pounded of the ratics of their Edes.