Book VI.

PROP. A. THEOR.

a 31. 1.

b 29. 1. c Hyp.

d 6. 1.

e 2. 6.

f 11. 5. 8 9. 5.

h 5. 1.

IF the outward angle of a triangle, made by producing one of its sides, be divided into two equal angles, by a straight line which also cuts the base produced; the segments between the dividing line and the extremities of the base have the same ratio which the other sides of the triangle have to one another: and if the segments of the base produced, have the same ratio which the other sides of the triangle have, the straight line drawn from the vertex to the point of section divides the outward angle of the triangle into two equal angles.

Let the outward angle CAE of any triangle ABC be divided into two equal angles by the straight line AD which meets the base produced in D: BD is to DC as BA to AC.

Through C draw CF parallel to AD a: and because the straight line AC meets the parallels AD, FC, the angle ACF is equal to the alternate angle CADb: but CAD is equal to the angle DAE; therefore also DAE is equal to the angle ACF. Again, because the straight line FAE meets the parallels AD, FC, the outward angle DAE is equal

A

F

E

to the inward and opposite
angle CFA: but the angle
ACF has been proved equal
to the angle DAE; therefore
also the angle ACF is equal
to the angle CFA, and conse-
quently the side AF is equal
to the side ACd: and because
AD is parallel to FC, a side of the triangle BCF, BD is to DC as
BA to AF; but AF is equal to AC; as therefore BD is to DC,
so is BA to AC.

C

D

Let now BD be to DC as BA to AC, and join AD; the angle CAD is equal to the angle DAE.

The same construction being made, because BD is to DC as BA to AC; and that BD is also to DC as BA to AFf; therefore BA is to AC as BA to AF8; wherefore AC is equal to AFh, and the angle AFC equal to the angle ACF: but

the angle AFC is equal to the outward angle EAD, and the Book VI. angle ACF to the alternate angle CAD; therefore also EAD is equal to the angle CAD. Wherefore, if the outward, &c. Q. E. D.

PROP. IV. THEOR.

THE sides about the equal angles of equiangular triangles are proportionals; and those which are opposite to the equal angles are homologous sides, that is, are the antecedents or consequents of the ratios.

Let ABC, DCE be equiangular triangles, having the angle ABC equal to the angle DCE, and the angle ACB to the angle DEC, and consequently a the angle BAC equal to the angle a 32. 1. CDE. The sides about the equal angles of the triangles ABC, DCE are proportionals; and those are the homologous sides which are opposite to the equal angles.

Let the triangle DCE be placed, so that its side CE may be contiguous to BC, and in the same straight line with it: and because the angles ABC, ACB are together less than two right angles, ABC, and DEC, which is F equal to ACB, are also less than two right angles; wherefore BA, ED produced shall meet; let them be A produced and meet in the point F: and because the angle ABC is equal to the angle DCE, BF is parallel d to CD. Again, because the angle ACB is equal to the angle DEC, AC is parallel to FEd; therefore

B

b 17.1.

D

c 12. Ax.

1.

d 28. 1.

C

E

FACD is a parallelogram; and consequently AF is equal to CD, and AC to FDe: and because AC is parallel to FE, one ofe 34. 1. the sides of the triangle FBE, BA is to AF, as BC to CE f: but f 2. 6. AF is equal to CD; therefore 8, as BA to CD, so is BC to CE; g 7. 5. and alternately, as AB to BC, so is DC to CE b: again, because CD is parallel to BF, as BC to CE, so is FD to DEF: but FD is equal to AC; therefore, as BC to CE, so is AC to DE: and alternately, as BC to CA, so CE to ED: therefore, because it has been proved that AB is to BC, as DC to CE, and as BC to CA, so CE to ED, ex æqualih, BA is to AC, as CD to DE. Therefore, h 22. 5. the sides, &c. Q. E. D.

Book VI.

PROP. V. THEOR.

a 23. 1.

IF the sides of two triangles, about each of their angles, be proportionals, the triangles shall be equiangular, and have their equal angles opposite to the homologous sides.

Let the triangles ABC, DEF have their sides proportionals, so that AB is to BC, as DE to EF; and BC to CA, as EF to FD; and consequently, ex æquali, BA to AC, as ED to DF; the triangle ABC is equiangular to the triangle DEF, and their equal angles are opposite to the homologous sides, viz. the angle ABC equal to the angle DEF, and BCA to EFD, and also BAC to EDF.

D

At the points E, F, in the straight line EF, make a the angle FEG equal to the angle ABC, and the angle EFG equal to BCA; wherefore the remaining angle BAC is equal to the b 32. 1. remaining angle EGF, and the triangle ABC is therefore equiangular to the triangle GEF; and consequently they have their sides opposite to the equal angles proportionals. Wherefore, as AB to

c 4.6.

d 11. 5. e 9. 5.

f 8. 1.

g 4.1.

A

B

4

E

G

BC, so is GE to EF; but as AB to BC, so is DE to EF; therefore as DE to EF, sod GE to EF: therefore DE and GE have the same ratio to EF, and consequently are equale for the same reason, DF is equal to FG: and because in the triangles DEF, GEF, DE is equal to EG, and EF common, the two sides DE, EF are equal to the two GE, EF, and the base DF is equal to the base GF; therefore the angle DEF is equal to the angle GEF, and the other angles to the other angles which are subtended by the equal sides 8. Wherefore the angle DFE is equal to the an gle GFE, and EDF to EGF and because the angle DEF is equal to the angle GEF, and GEF to the angle ABC; therefore the angle ABC is equal to the angle DEF: for the same reason, the angle ACB is equal to the angle DFE, and the angle at A to the angle at D. Therefore the triangle ABC is equiangular to the triangle DEF. Wherefore, if the sides, &c. Q. E. D.

Book VI.

PROP. VI. THEOR.

IF two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular, and shall have those angles equal which are opposite to the homologous sides.

Let the triangles ABC, DEF have the angle BAC in the one equal to the angle EDF in the other, and the sides about those angles proportionals; that is, BA to AC, as ED to DF; the triangles ABC, DEF are equiangular, and have the angle ABC equal to the angle DEF, and ACB to DFE.

At the points D, F, in the straight line, DF, make a the angle a 23. 1. FDG equal to either of the angles BAC, EDF; and the angle

DFG equal to the angle ACB; A

wherefore the remaining angle at B is equal to the remaining one at Gb, and consequently the triangle ABC is equiangular to the triangle DGF; and therefore as BA to AC, so is GD to DF: but,

C

D

E

F

by the hypothesis, as BA to B AC, so is ED to DF; as therefore ED to DF, so is d GD to DF; d 11. 5. wherefore ED is equal e to DG; and DF is common to the two e 9. 5. triangles EDF, GDF: therefore the two sides ED, DF are equal to the two sides GD, DF; and the angle EDF is equal to the angle GDF; wherefore the base EF is equal to the base FG, f4. 1. and the triangle EDF to the triangle GDF, and the remaining angles to the remaining angles, each to each, which are subtended by the equal sides: therefore the angle DFG is equal to the angle DFE, and the angle at G to the angle at E: but the angle DFG is equal to the angle ACB; therefore the angle ACB is equal to the angle DFE: and the angle BAC is equal to the angle EDF; wherefore also the remaining angle at B is equal to 8 Hyp the remaining angle at E. Therefore the triangle ABC is equiangular to the triangle DEF. Wherefore, if two triangles, &c. Q. E, D,

Book VI.

PROP. VII. THEOR.

See N.

IF two triangles have one angle of the one equal to one angle of the other, and the sides about two other angles proportionals, then, if each of the remaining angles be either less, or not less, than a right angle; or if one of them be a right angle: the triangles shall be equiangular, and have those angles equal about which the sides are proportionals.

Let the two triangles ABC, DEF have one angle in the one equal to one angle in the other, viz. the angle BAC to the angle EDF, and the sides about two other angles ABC, DEF propor tionals, so that AB is to BC, as DE to EF; and, in the first case, let each of the remaining angles at C, F be less than a right angle. The triangle ABC is equiangular to the triangle DEF, viz. the angle ABC is equal to the angle DEF, and the remaining angle at C to the remaining angle at F.

For, if the angles ABC, DEF be not equal, one of them is greater than the other: let ABC be the greater, and at the point B, in the straight line

AB, make the angle ABG a 23. 1. equal to the angle DEF: and because the angle at A is equal to the angle at D, and the angle ABG to the angle DEF; the remaining B b 32.1. angle AGB is equal to the

c 4. 6.

d 11. 5. e 9, 5.

f 5. 1.

remaining angle DFE: therefore the triangle ABG is equiangular to the triangle DEF; wherefore as AB is to BG, so is DE to EF; but as DE to EF, so, by hypothesis, is AB to BC; therefore as AB to BC, so is AB to BG d; and because AB has the same ratio to each of the lines BC, BG; BC is equal to BG, and therefore the angle BGC is equal to the angle BCG f: but the angle BCG is, by hypothesis, less than a right angle; therefore also the angle BGC is less than a right angle, and the adjacent angle AGB must be greater than a right angles. But it was proved that the angle AGB is equal to the angle at F; therefore the angle at F is greater than a right angle: but by the hypothesis, it is less than a right angle; which is absurd. There