parallelograms KC, EG are equiangular; therefore, as was shown in prop. 67, the parallelogram KC, that is, AC, is to EG, as MO to Q.

G D H

COR. 1. If two triangles, ABC, DEF have two equal angles, or two unequal, but given angles, ABC, DEF, and if the ratios of the sides about these angles, viz. the ratios of AB to DE, and of BC to EF be given; the triangles shall have a given ratio to one another.

B

A

CE F

Complete the parallelograms BG, EH: the ratio of BG to EH is given (67. or 68. dat.); and therefore the triangles which are the halves (34. 1.) of them have a given (15. 5. 72.) ratio to one another.

K A

L D

COR. 2. If the bases BC, EF of two triangles ABC, DEF have a given ratio to one another, and if also the straight lines AG, DH which are drawn to the bases from the opposite angles, either in equal angles, or unequal, but given angles, AGC, DHF have a given ratio to one another; the triangles shall have a given ratio to one another. Draw BK, EL parallel to AG, DH, and complete the parallelograms KC, LF. And because the angles AGC, DHF, or their equals, the angles KBC, LEF are either equal, or unequal, but given; and that the ratio of AG to DH, that is, of KB to LE, is given, as also the ratio of BC to EF; therefore (67. or 68. dat.) the ratio of the parallelogram KC to LF is given; wherefore also the ratio of the triangle ABC to DEF is given (41. 1. 15. 5.).

B G

E H F

Ir a parallelogram which has a given angle be applied to one side of a rectilineal figure given in species; if the figure have a given ratio to the parallelogram, the parallelogram is given in species.

Let ABCD be a rectilineal figure given in species, and to one side of it AB, let the parallelogram ABEF, having the given angle ABE, be applied; if the figure ABCD have a given ratio to the parallelogram BF, the parallelogram BF is given in species.

Through the point A draw AG parallel to BC, and through the point C draw CG parallel to AB, and produce GA, CB to the points H, K: because the angle ABC is given (3. def.), and the ratio of AB to BC is given, the figure ABCD being given in species; therefore the parallelogram BG is given (3. def.) in species. And because upon the same straight line AB the two rectilineal figures BD, BG given in species are described, the ra tio of BD to BG is given (53. dat.); and, by hypothesis, the ratio of BD to the parallelogram BF is given; wherefore (9. dat.) the ratio of BF, that is, (35. 1.) of the parallelogram BH, to BG is given, and therefore (1. 6.) the ratio of the straight line KB to BC is given; and the ratio of BC to BA is given, wherefore the ratio of KB to BA is given (9. dat.): and because the angle ABC is given, the adjacent angle ABK is given; and the angle ABE is given, therefore the remaining angle KBE is given. The angle EKB is also given, because it is equal to the angle ABK; therefore the triangle BKE is given in species, and consequently the ratio of EB to BK is given; and the ratio of KB to BA is given, wherefore (9. dat.) the ratio of EB to BA is given; and the angle ABE is given, therefore the parallelogram BF is given in species.

A

D

L

N

G

C

MO

B

P

ele

R

HF KE

A parallelogram similar to BF may be found thus: take a straight line LM given in position and magnitude; and because the angles ABK, ABE are given, make the angle NLM equal to ABK, and the angle NLO equal to ABE. And because the ratio of BF to BD is given, make the ratio of LM to P the same with it; and because the ratio of the figure BD to BG is given, find this ratio by the 53d dat. and make the ratio of P to Q the same. Also, because the ratio of CB to BA is given, make the ratio of Q to R the same; and take LN equal to R; through the point M draw OM parallel to LN, and complete the parallelogram NLOS; then this is similar to the parallelogram

BF.

Because the angle ABK is equal to NLM, and the angle ABE to NLO, the angle KBE is equal to MLO; and the angles BKE, LMO are equal, because the angle ABK is equal to NLM; therefore the triangles BKE, LMO are equiangular to one another; wherefore as BE to BK, so is LO to LM; and because as the figure BF to BD, so is the straight line LM to P; and as BD to BG, so is P to Q; ex æquali, as BF, that is (35.

1.) BH to BG, so is LM to Q: but BH is to (1. 6.) BG, as KB to BC: as therefore KB to BC, so is LM to Q; and because BE is to BK, as LO to LM; and as BK to BC, so is LM to Q and as BC to BA, so Q was made to R; therefore, ex æquali, as BE to BA, so is LO to R, that is to LN; and the angles ABE, NLO are equal; therefore the parallelogram BF is similar to LS.

Ir two straight lines have a given ratio to one another, and upon one of them be described a rectilineal figure given in species, and upon the other a parallelogram having a given angle; if the figure have a given ratio to the parallelogram, the parallelogram is given in species.*

Let the two straight lines AB, CD have a given ratio to one another, and upon AB let the figure AEB given in species be described, and upon CD the parallelogram DF having the given angle FCD; if the ratio of AEB to DF be given, the parallelogram DF is given in species.

E

F

B

Upon the straight line AB, conceive the parallelogram AG to be described similar, and similarly placed to FD; and because the ratio of AB to CD is given, and upon them are described the similar rectilineal figures AG, FD; the ratio of AG to FD is given (54. dat.); and the ratio of FD to AEB A is given; therefore (9. dat.) the ratio of AEB to AG is given; and the angle ABG is given, because it is equal to the angle FCD: because therefore the parallelogram AG which has a given angle ABG is

M

N

HK L

D

applied to a side AB of the figure AEB given in species, and the ratio of AEB to AG is given, the parallelogram AG is given (69. dat.) in species; but FD is similar to AG; therefore FD is given in species.

A parallelogram similar to FD may be found thus: take a straight line H given in magnitude; and because the ratio of the figure AEB to FD is given, make the ratio of H to K the same with it also, because the ratio of the straight line CD to AB is given, find by the 54th dat. the ratio which the figure FD described upon CD has to the figure AG described upon AB similar to FD; and make the ratio of K to L the same with this ratio;

* See Note.

and because the ratios of H to K, and of K to L are given, the ratio of H to L is given (9. dat.); because, therefore, as AEB to FD, so is H to K; and as FD to AG, so is K to L; ex æquali, as AEB to AG, so is H to L; therefore the ratio of AEB to AG is given; and the figure AEB is given in species, and to its side AB the parallelogram AG is applied in the given angle ABG; therefore by the 69th dat. a parallelogram may be found similar to AG: let this be the parallelogram MN; MN also is similar to FD; for, by the construction, MN is similar to AG, and AG is similar to FD; therefore the parallelogram FD is similar to MN.

If the extremes of three proportional straight lines have given ratios to the extremes of other three proportional straight lines; the means shall also have a given ratio to one another and if one extreme have a given ratio to one extreme, and the mean to the mean; likewise the other extreme shall have to the other a given ratio.

Let A, B, C be three proportional straight lines, and D, E, F three other; and let the ratios of A to D, and of C to F be given; then the ratio of B to E is also given.

Because the ratio of A to D, as also of C to F is given, the ratio of the rectangle A, C to the rectangle D, F is given (67. dat.); but the square of B is equal (17. 6.) to the rectangle A, C ; and the square of E to the rectangle (17. 6.) D, F; therefore the ratio of the square of B to the square of E is given; wherefore (58. dat.) also the ratio of the straight line B to E is given. Next, let the ratio of A to D, and of B to E be given; then the ratio of C to F is also given.

Because the ratio of B to E is given, the ratio of the square of B to the square of E is given (54. dat.); therefore (17. 6.) the ratio of the rectangle A, C to the rectangle D, F is given; and the ratio of the side A to the side D is given; therefore the ratio of the other side C to the other F is given (65. dat.).

A

D

E F

COR. And if the extremes of four proportionals have to the extremes of four other proportionals given ratios, and one of the means a given ratio to one of the means; the other mean shall have a given ratio to the other mean, as may be shown in the same manner as in the foregoing proposition.

IF four straight lines be proportionals; as the first is to the straight line to which the second has a given ratio, so is the third to a straight line to which the fourth has a giv

en ratio.

Let A, B, C, D be four proportional straight lines, viz. as A to B, so C to D; as A is to the straight line to which B has a given ratio, so is C to a straight line to which D has a given

ratio.

Let E be the straight line to which B has a given ratio, and as B to E, so make D to F: the ratio of B to E is given (Hyp.), and therefore the ratio of D to F; and because as A to B, so is C to D; and as B to E, so D to F; therefore, ex æquali, as A to E, so is C to F; and E is the straight line to which B has a given ratio, and F that to which D has a given ratio; therefore as A is to the straight line to which B has a given ratio, so is C to a line to which D has a given ratio.

PROP. LXXIII.

IF four straight lines be proportionals; as the first is to the straight line to which the second has a given ratio, so is a straight line to which the third has a given ratio to the fourth.*

Let the straight line A be to B, as C to D: as A to the straight line to which B has a given ratio, so is a straight line to which C has a given ratio to D.

Let E be the straight line to which B has a given ratio, and as B to E, so make F to C; because the ratio of B to E is given, the ratio of C to F is given and because A is to B, as C to D; and as B to E, so F to C; therefore, ex æquali in proportione perturbata (23. 5.), A is to E, as F to D; that is, A is to E, to which B has a given ratio, as F, to which C has a given ratio, is to D.

* See Note.

A

F

E

C D