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base bc, the same multiple is the triangle a h c of the triangle a bc. For the same reason, whatever multiple the base lc is of the base cd, the same multiple is the triangle a lc of the triangle a dc: and if the base hc be equal to the base cl, the triangle a hc is also equal to the triangle alc (i. 38) : and if the base hc be greater than the base c1, likewise the triangle a hc is greater than the triangle alc; and if less, less : therefore, since there are four magnitudes, viz. the two bases b c, cd, and the two triangles a b c, a cd; and of the base b c, and the triangle a b c, the first and third, any equimultiples whatever have been taken, viz. the base hc

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k and triangle a hc; and of the base cd and triangle a cd, the second and fourth, have been taken any equimultiples whatever, viz. the base cl and triangle ale; and that it has been shewn, that if the base hc be greater than the base cl, the triangle a hc is greater than the triangle alc; and if equal, equal; and if less, less : therefore (v. def. 5), as the base bc is to the base cd, so is the triangle a b c to the triangle a cd.

And because the parallelogram ce is double of the triangle a b c (i. 41), and the parallelogram cf double of the triangle a cd, and that magnitudes have the same ratio which their equimultiples have (v. 15); as the triangle a b c is to the triangle a cd, so is the parallelogram e c to the parallelogram cf; and because it has been shewn, that as the base bc is to the base cd, so is the triangle a b c to the triangle acd; and as the triangle a b c is to the triangle a cd, so is the parallelogram e c to the parallelogram cf; therefore, as the base bc is to the base cd, so is (v. 11) the parallelogram ec to the parallelogram cf. Wherefore, triangles, &c.' Q. Ê. D.

COR. From this it is plain, that triangles and parallelograms that have equal altitudes are one to another as their bases.

Let the figures be placed so as to have their bases in the same straight line ; and having drawn perpendiculars from the vertices of the triangles to the bases, the straight line which joins the vertices is parallel to that in which their bases are (i. 33), because the perpendiculars are both equal and parallel to one another. Then, if the same construction be made as in the proposition, the demonstration will be the same.

PROPOSITION II.—THEOREM. If a straight line be drawn parallel to one of the sides of a triangle, it shall

cut the other sides, or these produced, proportionally, and if the sides, or the sides produced, be cut proportionally, the straight line which joins the

points of section shall be parallel to the remaining side of the triangle. LET de be drawn parallel to bc, one of the sides of the triangle a bc:bd is to da, as ce to ea.

Join be, cd; then the triangle bd e is equal to the triangle cde (i. 37), because they are on the same base de, and between the same parallels de bc. But a de is another triangle, and equal magnitudes have to the same the same ratio (v. 7); therefore, as the triangle bde to the triangle a de, so is the triangle cde to the triangle a de, but as the triangle b de to the triangle a de, so is (vi. 1) bd to da, because having the same altitude, viz. the perpendicular drawn from the point e to a b,

d

a

a

a)

d

e

b

b

d

e b they are to one another as their bases; and for the same reason, as the triangle cde to the triangle a de, so is ce to ea. Therefore, as b d to d a, so is ce to e a (v. 11).

Next, let the sides a b, a c, of the triangle a b c, or these produced, be cut proportionally in the points d, e, that is, so that bd be to da as ce to ea, and join de; de is parallel to b c.

The same construction being made, because as b d to d a, so is ce to ea; and as b d to d a, so is the triangle b de to the triangle a de (vi. 1); and as ce to ea, so is the triangle cde to the triangle a de; therefore the triangle bde is to the triangle ad e, as the triangle cde to the triangle a de; that is, the triangles b de, cd e have the same ratio to the triangle ade; and therefore (v. 9) the triangle b d e is equal to the triangle cde; and they are on the same base de : but equal triangles on the same base are between the same parallels (i. 39); therefore de is parallel to bc. Wherefore, if a straight line, &c. Q. É. D.

PROPOSITION III.-THEOREM. If the angle of a triangle be divided into two equal angles by a straight line

which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another; and if the segments of the base have the same ratio which the other sides of the triangle

e

a

have to one another, the straight line drawn from the vertex to the point

of section divides the vertical angle into two equal angles. Let the angle bac of any triangle a b c be divided into two equal angles by the straight line ad : bd is to dc as b a to a C.

Through the point c draw ce parallel (i. 31) to da, and let ba produced meet ce in e; because the straight line ac meets the parallels a d, ec, the angle a ce is equal to the alternate angle cad (i. 29): but cad, by the hypothesis, is equal to the angle bad; wherefore bad is equal to the angle ace. Again, because the straight line ba e meets the parallels a d, ec, the outward angle bad is equal to the inward and opposite angle a ec: but the angle a ce has been proved equal to the angle bad; therefore also a ce is equal to the angle a ec, and consequently the b

d side a e is equal to the side a c (i. 6): and because a d is drawn parallel to one of the sides of the triangle bce, viz. to ec, bd is to dc, as ba to a e (vi. 2), but ae is equal to ac; therefore as bd to dc, so is ba to a c (v. 7).

Let now bd be to dc, as b a to a c, and join a d; the angle bac is divided into two equal angles by the straight line a d.

The same construction being made ; because, as bd to d c, so is b a to ac; and as bd to cd so is b a to a e (vi. 2), because a d is parallel to ec; therefore ba is to a c, as ba to a e (v. 11); consequently a c is equal to a e (v. 9), and the angle a ec is therefore equal to the angle a ce (i: 5): but the angle a ec is equal to the outward and opposite angle bad; and the angle a ce is equal to the alternate angle cad (i. 29) : wherefore also the angle bad is equal to the angle cad; therefore the angle bac is cut into two equal angles by the straight line ad. Therefore, if the angle, &c. Q. E. D.

PROPOSITION A.—THEOREM.

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 thvertex to the point of section divides the outward angle of the triangle into

two equal angles. LET the outward angle ca e of any triangle abc be divided into two qual angles by the straight line ad which meets the base produced in : bd is to dc, as ba to a c.

Through c draw cf parallel to ad (i. 31); and because the straight line a c meets the parallels a d, cf, the angle a cf is equal to the alternate

a

angle cad (i. 29): but cad is equal to the angle da e (hyp.); therefore also da e is equal to the angle a cf. Again, because the straight line fa e

meets the parallels a d, fc, the outward e

angle da e is equal to the inward and opposite angle cfa: but the angle a cf has been proved equal to the angle dae; therefore also the angle a cf is equal to the angle cfa, and conse

quently the side af is equal to the side b с

d ac (i. 6): and because a d is parallel

to fc, a side of the triangle bcf, bd is to dc, as ba to a f (vi. 2); but a f is equal to ac; as therefore bd is to dc, so is ba to a c.

Let now bd be to dc as b a to a c, and join ad; the angle cad is equal to the angle da e.

The same construction being made, because bd is to dc as b a to ac; and that bd is also to dc, as ba to a f (vi. 2); therefore ba is to a c as ba to af (v. 11); wherefore a c is equal to a f (v. 9), and the angle afc equal (i. 5) to the angle a cf: but the angle a fc is equal to the outward angle è a d, and the angle a cfto the alternate angle cad; therefore also ead is equal to the angle cad. Wherefore, if the outward, &c. Q. E. D.

PROPOSITION IV.—THEOREM. 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 a bc, dce be equiangular triangles, having the angle a b c equal to the angle dce, and the angle a cb to the angle de c, and consequently (i. 32) the angle bac equal to the angle cde. The sides about the equal angles of the triangles a b c, 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 b c, and in the same straight line with it: and because the angles a b c,

acb, are together less than two right angles (i. 17), a b c and dec, which is equal to a cb, are also less than two right angles ; wherefore ba, ed produced shall meet (i. ax. 12); let

them be produced and meet in the point f; a d

and because the angle abc is equal to the angle dce, bf is parallel to cd (i. 28). Again, because the angle a cb is equal to the angle

dec, ac is parallel to fe (i. 28): therefore e

facd is a parallelogram, and consequently af

is equal to cd, and ac to fd (i. 34): and beca use a c is parallel to fe, one of the sides of the triangle fb e, ba is to af as b c to ce (vi. 2): but a f is equal to cd; therefore (v. 7), as ba to cd, so is bc to ce; and alternately, as a b to bc, so is dc to ce.

b

Again, because cd is parallel to bf, as b c to ce, so is fd to de (vi. 2); but fd is equal to ac; therefore, as b c to ce, so is ac to de: and alternately, as b c to ca, so is ce to ed: therefore, because it has been proved that a b is to b c, as dc to ce, and as b c to ca, so is ce to ed, ex æquali (v. 22), ba is to ac, as cd to de. Therefore the sides, &c. Q. E. D.

PROPOSITION V.—THEOREM.

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 a b c d e f have their sides proportionals, so that a b is to bc as de to ef; and b c to ca as efto fd; and consequently, ex æquali, ba to a c as ed to df; the triangle a bc is equiangular to the triangle d e f, and their equal angles are opposite to the homologous sides, viz. the angle a b c equal to the angle def, and bca to efd, and also bac to edf.

At the points e, f, in the straight line ef, make (i. 23) the angle feg equal to the angle a b c, and the angle efg equal to bca; wherefore the remaining angle bac is equal to the remaining angle egf (i. 32), and the triangle abc is therefore equiangular to the triangle

d gef; and consequently they have their sides opposite to the equal angles proportionals (vi. 4). Wherefore as a b to bc, so is

f ge to ef; but as a b to bc, so is de to ef; therefore as de to ef, so is ge to ef b (v. 11): therefore de and

ge

have the same ratio to ef, and consequently are equal (v. 9): for the same reason, d f 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 dfis equal to the base gf; therefore the angle def is equal (i. 8) to the angle gef, and the other angles to the other angles which are subtended by the equal sides (i. 4). Wherefore the angle d fe is equal to the angle gfe, and edf toegf: and because the angle def is equal to the angle g ef, and gef to the angle a bc; therefore the angle a b c is equal to the angle def. For the same reason, the angle a cb is equal to the angle dfe, and the angle at a to the angle at d. Therefore the triangle a bc is equiangular to the triangle def. Wherefore, if the sides, &c. Q. E. D.

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PROPOSITION VI.—THEOREM.

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

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