base bc, the same multiple is the triangle a hc 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 ale 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 h c be greater than the base cl, 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 bc, cd, and the two triangles abc, acd; and of the base bc, and the triangle a b c, the first and third, any equimultiples whatever have been taken, viz. the base hc

a

:

h g b c

d

k

1

and triangle a h c; and of the base c d and triangle a c d, 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 he be greater than the base c 1, 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 abc to the triangle a cd.

And because the parallelogram ce is double of the triangle a bc (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 abc is to the triangle a cd, so is the parallelogram ec to the parallelogram cf; and because it has been shewn, that as the base bc is to the base cd, so is the triangle abc to the triangle a cd; and as the triangle abc 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. E. 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 be: bd is to da, as ce to ea.

Join be, cd; then the triangle bde 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 bde 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,

they are to one another as their bases;

e b

C

and for the same reason, as the triangle cde to the triangle ade, so is ce to ea. Therefore, as bd to da, so is ce to ea (v. 11).

Next, let the sides a b, a c, of the triangle a bc, 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 bc.

The same construction being made, because as bd to da, so is ce to ea; and as bd to da, so is the triangle bde 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 a de, as the triangle c d e to the triangle a de; that is, the triangles bde, cde have the same ratio to the triangle ade; and therefore (v. 9) the triangle bde 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.

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

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 ba to a c.

e

a

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 ad, ec, the angle ace 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 bae meets the parallels a d, ec, the outward angle bad is equal to the inward and opposite angle aec: but the angle a ce has been proved equal to the angle bad; therefore also a ce is equal to the angle aec, and consequently the b side a e is equal to the side a c (i. 6):

d

and because ad is drawn parallel to one of the sides of the triangle bce, viz. to ec, bd is to dc, as ba to ae (vi. 2), but ae is equal to ac; therefore as bd to dc, so is ba to ac (v. 7).

Let now bd be to dc, as ba 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 ba to ac; and as bd to cd so is ba to ae (vi. 2), because ad is parallel to ec; therefore ba is to a c, as ba to ae (v. 11); consequently ac is equal to a e (v. 9), and the angle aec is therefore equal to the angle ace (i. 5) but the angle a ec is equal to the outward and opposite angle bad; and the angle ace 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 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 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 ac meets the parallels a d, cf, the angle a cf is equal to the alternate

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

b

C

e

meets the parallels a d, fc, the outward angle dae 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 acf is equal to the angle cfa, and consequently the side af is equal to the side dac (i. 6) and because ad is parallel to fc, a side of the triangle bcf, bd is

to dc, as ba to af (vi. 2); but af is equal to a c; as therefore bd is to dc, so is ba to a c.

Let now bd be to dc as ba 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 ba to ac; and that bd is also to dc, as ba to a f (vi. 2); therefore ba is to ac as ba to af (v. 11); wherefore a c is equal to af (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 e a d, and the angle a c f to 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 abc, dce be equiangular triangles, having the angle abc equal to the angle dce, and the angle acb 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 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,

a

d

a cb, are together less than two right angles (i. 17), a b c and d e c, 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; and because the angle abc is equal to the angle dce, bf is parallel to c d (i. 28). Again, because the angle a cb is equal to the angle dec, ac is parallel to fe (i. 28): therefore facd is a parallelogram, and consequently af is equal to cd, and ac to fd (i. 34): and because a c is parallel to fe, one of the sides of the triangle fbe, ba is to afas bc to ce (vi. 2): but af is equal to cd; therefore (v. 7), as ba to cd, so is be to ce; and alternately, as ab to bc, so is dc to ce

b

c

e

Again, because cd is parallel to bf, as bc to ce, so is fd to de (vi. 2) ; but fd is equal to ac; therefore, as bc to ce, so is a c to de: and alternately, as bc to ca, so is ce to ed: therefore, because it has been proved that ab is to bc, as dc to ce, and as bc to ca, so is ce to ed, ex æquali (v. 22), ba is to a c, 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 abc, def have their sides proportionals, so that ab is to be as de to ef; and bc to ca as efto fd; and consequently, ex aequali, 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.

a

At the points e, f, in the straight line ef, make (i. 23) the angle feg equal to the angle a bc, 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 gef; and consequently they have their sides opposite to the equal angles proportionals (vi. 4). Wherefore as ab to bc, so is ge to ef; but as a b to bc, so is de to ef; therefore as de to ef, so is ge to ef (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

b

مع

g

3

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 dfe is equal to the angle 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 a bc 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 abc is equiangular to the triangle def. Wherefore, if the sides, &c. Q. E. D.

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