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Book VI. wherefore as A to B, fo B to C. Therefore if three ftraight lines, &c. Q. E. D.

I.

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UPON

PROP. XVIII. PROB.

PON a given ftraight line to defcribe a rectilineal figure fimilar, and fimilarly fituated to a given rectilineal figure.

Let AB be the given ftraight line, and CDEF the given rectilineal figure of four fides; it is required upon the given straight line AB to defcribe a rectilineal figure fimilar and fimilarly 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 ABG equal to the angle CDF; therefore the remaining angle CFD is equal to the b. 32. 1. remaining angle AGB . wherefore the triangle FCD is equiangular to the triangle GAB.

C. 4. 6.

d. 22. 5.

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the remaining angle GHB, and the triangle FDE equiangular to the triangle GBH. then becaufe 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 fame reafon, the angle ABH is equal to the angle CDE; alfo the angle at A is equal to the angle at C, and the angle GHB to FED. therefore the rectilineal figure ABHG is equiangular to CDEF. but likewife thefe figures have their fides about the equal angles proportionals. because the triangles GAB, FCD being equiangular, BA is to AG, as DC to CF; and becaufe AG is to GB, as CF to FD; and as GB to GH, fo, by reafon of the equiangular triangles BGH, DFE, is FD to FE; therefore, ex aequali, AG is to GH, as CF to FE. in the fame manner it may be proved that AB is to BH, as CD to DE. and GH is to HB, as FE to ED c. Wherefore because the rectilineal

figures ABHG, CDEF are equiangular, and have their fides about Book VI. the equal angles proportionals, they are fimilar to one another .

Next, Let it be required to defcribe upon a given ftraight line c.1.Def.4. AB, a rectilineal figure fimilar, and fimilarly fituated to the rectilineal figure CDKEF of five fides.

Join DE, and upon the given ftraight line AB defcribe the rectilineal figure ABHG fimilar and fimilarly fituated to the quadrilateral figure CDEF, by the former cafe. 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 remaining angle at L. and becaufe the figures ABHG, CDEF are fimilar, 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 fame reafon, the angle ABL is equal to the angle CDK. therefore the five fided figures AGHLB, CFEKD are equiangular. and because the figures AGHB, CFED are fimilar, GH is to HB, as FE to ED; and as HB to HL, fo is ED to EK; therefore ex c. 4. 6. aequali, GH is to HL, as FE to EK. for the fame reafon, AB is d. 22. 5, 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 fided figures AGHLB, CFEKI are equiangular, and have their fides about the equal angles proportionals, they are fimilar to one another. and in the fame manner a rectilineal figure of fix fides may be defcribed upon a given ftraight line fimilar to one given, and fo on. Which was to be done.

SIN

PROP. XIX. THE OR,

IMILAR triangles are to one another in the du plicate ratio of their homologous fides.

Let ABC, DEF be fimilar triangles having the angle B equal to the angle E, and let AB be to BC, as DE to EF, fo that the fide

BC is homologous to EF, the triangle ABC has to the triangle a.12,Def.§. DEF, the duplicate ratio of that which BC has to EF,

Take BG a third proportional to BC, EF b, fo that BC is to EF, b, 11. 6, as EF to BG, and join GA. then, because as AB to BC, fo DE to

EF; alternately, AB is to DE, as BC to EF. but as BC to EF, fo c, 10. 5.

Book VI. is EF to BG; therefore d as AB to DE, fo is EF to BG. wherew fore the fides of the triangles ABG, DEF which are about the equal angles are reciprocally proportional. but triangles which have the fides about two equal angles reciprocally proportional are equal to one another ". there

d. 11. 5.

f. 15.6.

fore the triangle ABG
is equal to the triangle
DEF. and because as
BC is to EF, fo EF to
BG; and that if three

ftraight lines be propor

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f.10.Def.5. tionals, the firft is faid B G

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to have to the third the

duplicate ratio of that which it has to the fecond; BC therefore has to BG the duplicate ratio of that which BC has to EF. but as BC to BG, fo 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; wherefore alio 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 ftraight lines be proportionals, as the first is to the third, fo is any triangle upon the first to a fimilar and fimilarly defcribed triangle upon the fecond.

PROP. XX. THEOR.

SIMILAR polygons may be divided into the fame number of fimilar triangles, having the fame ratio 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 fimilar polygons, and let AB be the homologous fide to FG. the polygons ABCDE, FGHKL may be divided into the fame number of fimilar triangles, whereof each to each has the fame ratio which the polygons have; and the polygon ABCDE has to the polygon FGHKL the duplicate ratio of that which the fide AB has to the fide FG.

Join BE, EC, GL, LH. and because the polygon ABCDE is Book 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 an 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 fimilar, 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 fimilar, EB is to BA, as LG to GF; and alfo, because the polygons are fimilar, AB is to BC, as FG to GH; therefore, ex aequali, EB is to BC, as LG to GH; that is, the fides about d. 22. 5, the equal angles EBC, LGH are proportionals; therefore the

a

triangle EBC is equiangular to the triangle LGH, and fimilar to

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wided into the fame number of fimilar triangles.

Alfo thefe triangles have, each to each, the fame ratio which the polygons have to one another, the antecedents being ABE, EBC, ECD, and the confequents FGL, LGH, LHK. and the polygon ABCDE has to the polygon FGHKL the duplicate ratio of that which the fide AB has to the homologous fide FG.

C. 19.6.

Because the triangle ABE is fimilar to the triangle FGL, ABE has to FGL the duplicate ratio of that which the fide BE has to the fide GL. for the fame reason, the triangle BEC has to GLH the duplicate ratio of that which BE has to GL. therefore as the triangle ABE to the triangle FGL, fof is the triangle BEC to the triangle f. 11. § GLH. Again, because the triangle EBC is fimilar to the triangle LGH, EBC has to LGH, the duplicate ratio of that which the fide EC has to the fide LH. for the fame reason, the triangle ECD has to the triangle LHK, the duplicate ratio of that which EC has to LII, as therefore the triangle EBC to the triangle LGH, fo is f the

Book VI. triangle ECD to the triangle LHK. but it has been proved that the triangle EBC is likewife to the triangle LGH, as the triangle ABE to the triangle FGL. Therefore as the triangle ABE is to the triangle FGL, fo is triangle EBC to triangle LGH, and triangle ECD to

triangle LHK.

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and therefore

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dents to all the

B. 11. 5. confequents &

h.10.Def.5.

Wherefore as the triangle ABE to the triangle FGL, fo 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 alfo the polygon ABCDE has to the polygon FGHKL the duplicate ratio of that which AB has to the homologous fide FG, Wherefore fimilar polygons, &c. Q. E. D.

COR. I. In like manner it may be proved that fimilar four fided 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 univerfaliy, fimilar rectilineal fi gures are to one another in the duplicate ratio of their homologous fides.

COR. 2. And if to AB, FG two of the homologous fides a third proportional M be taken, AB has to M the duplicate ratio of that which AB has to FG. but the four fided figure or polygon upon AB has to the four fided figure or polygon upon FG likewife 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 alfo proved in triangles . Therefore, univerfally, it is

manifeft, that if three ftraight lines be proportionals, as the first is to the third, fo is any rectilineal figure upon the first, to a fimilar and fimilarly defcribed rectilineal figure upon the fecond,

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