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* 32. 1. &

B

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 • 23. 1. ABG equal to the angle CDF; therefure the remaining angle CFD is equal

H

G to the remaining angle AGB*: there. fore the triangle FCD is equiangular

K K 3 Ax. to the triangle GAB : again, at the A points G, B, in the straight line GB, make * the angle BGH equal to the angle DFE, and the angle 23. 1. GBH equal to FDE; therefore the remaining angle FED is equal to the remaining angle GHB, and the triangle FDE equiangular to the triangle GBH: then, because the angle 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, + 2 Ax. the angle ABH is equal to the angle CDE: also the angle at A is equal t to the angle at C, and the angle GHB to FED; † Constr. therefore the rectilineal figure ABHG is equiangular to CDEF: 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, DFE, is FD to FE; therefore, ex æquali *, 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 * 4.6. ED. Wherefore, because the rectilineal figures ABHG, CDEF are equiangular, and have their sides about the equal angles proportionals, they are similar * to one another.

• 1 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 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; therefore the remaining angle at K is equal + to the † 32. 1. & remaining angle at L: and because the figures ABHG, CDEF are similar, the angle GHB is equal † to the angle FED: and + 1 Def. 6. BHL is equal to DEK; wherefore the whole angle GAL is equal to the whole anglc FEK: for the same reason, the angle

# 4. 6.

22. 5.

3 Ax.

t I Def. 6.

* 4. 6.

22. 5.

• 4. 6.

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; 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, 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.

PROPOSITION XIX.

THEOR.—Similar triangles are to one another in the duplicalc

ratio of their homologous sides.

16. 5.

A

B

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, $o * 12 Def. 5. 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. 11. 6. 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 + Constr.

to DE, as BC to EF: but as BC to EF +, * 11. 5.

D so is EF to BG; therefore *, as AB to DE, so is EF to BG: therefore the sides of the triangles ABG, DEF, which are G CE about the equal angles, are reciprocally proportional: but triangles, which have the sides about two equal angles reciprocally proportional, are equal* to one another; therefore the triangle ABG is equal to the triangle

DEF : and because as BC is to EF, so EF to BG, and that if * 10 Def. 5. three straight lines be proportional, the first is said * to have

to the third the 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 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; there

15. 6.

• 1.6.

fore 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.

PROPOSITION XX.

Tukor.—Similar polygons may be divided into the same num

ber 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 sides have.

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 • 1 Def. 6. angle GFL, and BA is to AE *, as GF to FL: therefore, because * 1 Def. 6. the triangles ABE, 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 . 6.6. FGL, and therefore * similar to it;

A wherefore the angle ABE is equal

M F

E to the angle FGL: and, because the

B

G polygons are similar, the whole

D С K H angle ABC is equal * to the whole

* 1 Def.6. angle FGH, therefore the remaining angle EBC is equal to the + 3 Ax. remaining angle LGH: and because the triangles ABE, FGL are similar, EB is to BA *, as LG to GF; and also, because the * 4.6. polygons are similar, AB is to BC *, as FG to GH; therefore, • 1 Def. 6. ex æquali *, EB is to BC, as LG to GH; that is, the sides 22. 5. about the equal angles EBC, LGH, are proportionals ; therefore the triangle EBC is equiangular to the triangle LGH, and • 6.6. similar * to it; for the same reason, the triangle ECD likewise • 4. 6. is similar to the triangle LHK: therefore, the similar polygons

• 4. 6.

19. 6.

* 11. 5.

ABCDE, FGHKL are divided into the same number of similar triangles.

Also these triangles shall 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 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 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 BE 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 LII: 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

M. F triangle LGH, as the triangle ABE

ES to the triangle FGL; therefore, as

B

G the triangle ABE to the triangle

с K II FGL, so is triangle EBC to triangle LGII, and triangle ECD to triangle LHK: and therefore, as one of the antecedents to one of the consequents *, so are all the antecedents to all the consequents; that is, 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 poly

11.5.

A

* 12. 5.

† 19, 6.

gons, &c.

Q. E. D.

† 19. 6.

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, a

third + proportional M be taken, AB • has to M the duplicate + 11. 6. ratio of that which AB has to FG: but the four-sided figure

10 Del.5. or polygon upon AB, has to the four-sided figure or polygon upon FG, likewise the duplicate ratio t of that which AB has † Cor. 1. to FG ; therefore t, as AB is to M, so is the figure upon AB † 11. 5. to the 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.

PROPOSITION 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 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 proportional: again, | Def. 6. because B is similar to C, they are equiangular, and have their sides about the equal angles proportionals: there

• Def. 6. fore 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 are equiangular, and have their • Ax. I. sides about the equal angles proportionals: therefore A is • 11.5. similar * to B. Therefore, rectilineal figures, &c. Q. E. D. • I Def. 6.

A

B

PROPOSITION 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

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