FD is to GB, fo is FC to GA and CD to AB: and therefore [by 11. 5.] as F C is to GA, fo is CD to A B and FE to GH, and moreover E D to H B. Wherefore because the angle CFD [by conftruction] is equal to the angle AGB, and the angle DFE to the angle BGH: the whole angle CF E is equal to the whole angle AGH. By the fame reason the angle CDE is alfo equal to the angle ABH, and befides the angle at c is equal to the angle at A; but the angle at E is equal to the angle at H; therefore AH is equiangular to CE, and it has the fides about the equal angles proportional: Wherefore [by 1. def. 6] the right lined figure A H will be fimilar to the right lined figure

CE.

Therefore upon the given right line A B the right lined figure AH is defcribed fimilar to the given right lined fi gure CE, and alike fituate. Which was to be done.

D

h Similar right lined figures are faid to be alike described, or fet upon right lines, when thofe right lines are the homologous fides of the fimilar right lined figures; or when the equal angles are conftituted upon thofe right lines, and the remaining equal angles, and proportional fides of the figures always follow one another in order As the triangles ABC, DEF are not only fimilar but are alike fituate upon the right lines BC, EF, when the angles B, C are equal to the angles A E, F; and it be as A B to BC, fo is DE to EF, &c. But they would not have been fimilarly fituate, if B the angle B had been equal to F, and c equal to E. ner the rectangles AC, EG are

In like man- A

CE

BIK

C

the right lines pc, HG, when
AD is to DC as EH is to HG E

F

H G

G

fimilar and alike fituate upon D

H

c. But the rectangles a C, IG are faid not to be fimilar fituate upon the right lines DC, HG although they are fimilar, as is manifelt. But notwithstanding the fame figures will be alike fituate or described upon the right lines DC, 1H, or upon

AD, HG.

That the figures be alike fituate, is a neceffary condition. For if they be not, this propofition, and all the o hers following, limited to that condition will not be always true, because two fimilar figures may be described upon the fame right line not

I alike fituate, that will be unequal, as the triangles A B C, DEF, (where AC is equal to DF, and the angles A, D ; B, E Fare refpectively equal to one another) will be unequal.

F

Note alfo that all equilateral and equiangular figures are alike defcribed upon any right lines.

PROP. XIX. THEOR.

Similar triangles are to one another in the duplicate ratio of their bomologous fides.

Let the trianges ABC, DEF be fimilar, having the angle B equal to the angle E; and let A B be to BC, as DE is to EE: fo that [by 12. def. 5.] the fide B C is homologous to the fide EF: I fay the triangle A B C is to the triangle DEF in the duplicate ratio of that of the fide BC to the fide E F.

For [by 11. 6.] find a third proportional BG to BC, EF, fo that BC be to EF, as EF is to BG; and join GA.

BG

CE

8

Then because it is as

A B to BC, fo is DE to EF; it will be alternately D [by 16. 5.] as A B is to DE, fo is BC to E F. But as BC is to E F, fò is E F to BG; and therefore [by II. 5.] as AB is to DE, fo is EF to BG: Therefore the fides about the equal angles of the triangles ABG, DEF, are reciprocally proportional. But thofe triangles having one angle of the one equal to one angle of the other, and the fides about the equal angles reciprocally proportional, are [by 15. 6.] equal to one another: therefore the triangle ABG is equal to the triangle DE F. And because B C is to E F, as E F is to B G; and if three right lines be proportional, the ratio of the firft to the third is [by 10. def. 5.] duplicate to the ratio that the firft has to the fecond: the ratio of BC to BG will be duplicate of the ratio of B C to E F. But as BC is to B G, fo [by 1. 6.] is the triangle ABC to the triangle ABG: Therefore also

the

the ratio of the triangle A B C to the triangle ABG is duplicate of the ratio of BC to EF. But the triangle A B G is equal to the triangle DEF: and therefore [by 7. 5.] the ratio of the triangle ABC to the triangle D E F is the duplicate of the ratio of BC to EF.

Therefore fimilar triangles are to one another in the duplicate ratio of their homologous fides. Which was to be demonftrated.

Corollary. From hence it is manifeft, if three right lines be proportional, as the first is to the third, fo is a triangle described upon the first, to a triangle fimilar and alike fituated, defcribed upon the second: because it has been proved as CB is to BG, fo is the triangle ABC to the triangle AB G, that is, to the triangle DEF.

Similar polygons are divided into equal numbers of fimilar triangles homologous to the wholes; and one polygon has to another polygon a duplicate ratio to that which one homologous fide of the one bas to an homologous fide of the other.

Let the polygons A B C DE, FGH KL be fimilar, and let A B, FG be homologous fides: I fay the polygons ABCDE, FG HKL are divided into an equal number of fimilar triangles, homologous to the wholes; and the polygon ABCDE to the polygon FGH KL has a duplicate ratio to that which one homologous fide A B has to another FG.

ing one angle of the one equal to one angle of the other, and the fides about the equal angles proportional: [by 6. 6.] the triangle ABE will be equiangular to the triangle FGL: Therefore [by 4. 6. it will be alfo fimilar to it; and fo the angle ABE is equal to, the angle FG L. But also the whole angle ABC is equal to the whole angle F G H, because the polygons are fimilar: Therefore the remaining angle EBC is equal to the remaining angle LGH. And because the triangles. A BE, FGL are fimilar, it is as EB is to BA, fo is LG to GF; and because of the fimilar polygons A B is to BC, as FG is to GH; it will be by equality, [by 22. 5.] as E B is to BC, fo is LG to GH, viz. the fides proportional about the equal angles : Therefore [by 6. 6.] the triangle EBC is equiangular to the triangle LGH; and also [by 4. 6.] fimilar to it. By the fame reason the triangle ECD is fimilar to the triangle LHK. Therefore the polygons ABCDE, FGH KL are divided into equal numbers of fimilar triangles.

I fay alfo the triangles are homologous to the wholepolygons: that is, the triangles are proportional, and the antecedents are ABE, EBC, ECD, and the confequents are FG L, LG H, LH K; and one polygon A B C D E has to the other F GHK L a ratio being the duplicate of the ratio that one homologous fide has to the other, that is of AB to FG.

For join A C, F H.

Then because the polygons are fimilar, the angle ABC is equal to the angle FG H, and it is as A B to B C, fo is F G to GH: [by 6. 6.] the triangle ABC will be equiangular to the triangle FGH; and fo the angle B A C is equal to the angle GFH, and the angle BCA equal to the angle GHF. Moreover, because the angle BAM is equal to the angle G F N, and it has been proved that the angle ABM is equal to the angle FGN; [by 32. 1.] the remaining angle AMB will be equal to the remaining angle FNG: Therefore the triangle A B M is equiangular to the triangle FGN. In like manner we prove that the triangle BMC is equiangular to the triangle GNH: Therefore [by 4. 6.] as AM is to ME, fo is FN to NG, and as BM is to MC, fo is GN to NH: Wherefore by equality [by 22. 5.] as AM is to M C, fo is F N to N H. But as A M is to MC, fo is the triangle A B M to the triangle M B C, and the triangle AME to the triangle EMC: For [by 1, 6.] they are

to

to one another as their bases: and [by 12. 5.] as one of the antecedents is to one of the confequents, fo are all the antecedents to all the confequents: Therefore as the triangle AMB is to the triangle B M C, fo is the triangle ABE to the triangle CBE. But as A M B is to B M C, so is AM to MC; and therefore [by 11. 5.] as AM is to MC, fo is the triangle ABE to the triangle EEC. By the fame reason, as FN is to N H, fo is the triangle FGL to the triangle GLH. But AM is to M C, as FN is to NH: Therefore [by 11. 5.] as the triangle ABE is to the triangle BEC, fo is the triangle FGL to the triangle GLH; and [by 16. 5.] as the triangle ABE is to the triangle FGL, fo is the triangle BEC to the triangle GLH. After the like manner we demonftrate, by joining BD, GK, that the triangle BEC is to the triangle GLH, as the triangle FCD is to the triangle L HK. And because the triangle ABE is to the triangle FGL, as the triangle EBC is to the triangle LGH, and fo is the triangle ECD to the triangle LHK; it will be [by 12. 5.] as one of the antecedents is to one of the confequents, so are all the antecedents to all the confequents: Therefore as the triangle ABE is to the triangle F GL, fo is the polygon ABCDE, to the polygon FGHKL. But the triangle ABE to the triangle FGL, has a ratio duplicate to that which the homologous fide A B has to the homologous fide FG: for [by 19. 6.] fimilar triangles are in the duplicate ratio of their homologous fides: therefore also the polygon ABCDE to the polygon FGHKL has a ratio duplicate to that which the homologous fide A B of the one, has to the homologous fide F G of the other.

Therefore fimilar polygons are [or may be] divided into equal numbers of fimilar triangles, homologous to the whole polygons; and one polygon to the other has a ratio, which is the duplicate of the ratio that one homologous fide of the one has to an homologous fide of the other. Which was to be demonftrated.

Corollary 1. We demonstrate after the fame manner in fimilar quadrilateral figures, that they are to one another in the duplicate ratio of their homologous fides. This has been alfo proved of triangles, therefore univerfally fimilar right lined figures are to one another in the duplicate ratio of their homologous fides.

Coral