20. 6.

fide of the figure to be described, homologous to BC the fide of D, and the figure itself can be described by the 18th prop.

b. 6. which, by the construction, is similar to D; and because 8 2. Cor. Dis to A, as & 'BC to CL, that is as the figure BK to KL; and

that D is equal to BK, therefore A is equal to KL, that is, to b 14. So

H. 57.

PRO P. LXI. See N.

IF a parallelogram given in magnitude has one of its

fides and one of its angles given in magnitude, the other side also is given.

Let the parallelogram ABDC given in magnitude, have the fide AB and the angle BAC given in magnitude, the other side AC is given.

Take a straight line EF given in position and magnitude ; and because the parallelogram AD A

B is given in magnitude, a rectilineal a 1. def. figure equal to it can be found 2.

And a parallelogram equal to this
Cor. 45. figure can be applied to the given C

D
straight line EF in an angle equal to
the given angle BAC. Let this be
the parallelogram EFHG having
the angle FEG equal to the angle
BAC. And because the parallelo-G H
grams AD, EH are equal, and have
the angles at A and E equal; the sides about them are recipro-
cally proportional; therefore as AB to EF, so is EG to AC;
and AB, EF, EG are given, therefore also AC is given 4.
Whence the way of finding AC is manifeft.

PRO P. LXII.
IF a parallelogram has a given angle, the rectangle con-

tained by the sides about that angle has a given ratio to the parallelogram.

Let the parallelogram ABCD have the A D given angle ABC, the rectangle AB, BC has a given ratio to the parallelogram AC.

From the point A draw AE perpendicular to BC; 'because the angle ABC is

C given, as also the angle AEB, the triangle 43. dat. ABE is given in species ; therefore the

ratio of BA O AE is given. But as BA b 1. 6. to AE, fo is the rectangle AB, BC to the

GK H rectangle AE, BC; therefore the ratio of

che

€ 14. 6.

H.

Sec N.

E
FI

the rectangle AB, BC to AE, BC, that is, to the parallelo- c 35. r. gram AC, is given.

And it is evident how the ratio of the rectangle to the parallelogram may be found, by making the angle FGH equal to the given angle ABC, and drawing, from any point F in one of its fides, FK perpendicular to the other GH; for GF is to FK, as BA to AĒ, that is, as the rectangle AB, BC, to the parallelogram AC. Cor. And if a triangle ABC has a given angle ABC, the

66. rectangle AB, BC contained by the fides about that angle, shall have a given ratio to the triangle ABC.

Complete the parallelogram ABCD; therefore, by this proposition, the rectangle AB, BC has a given ratio to the paral. lelogram AC; and AC has a given ratio to its half the triangled ABC, therefore the rectangle AB, BC has a given raad 14. 1: tio to the triangle ABC.

And the ratio of the rectangle to the triangle is found thus: Make the triangle FGK, as was shown in the proposition ; the ratio of GF to the half of the perpendicular FK is the same with the ratio of the rectangle AB, BC to the triangle ABC. Be. cause, as was shown, GF is to FK, as AB, BC to the parallelogram AC; and FK is to its half, as AC is to its balf, which is the triangle ABC, therefore, ex aequali, GF is to the half of FK, as AB, BC rectangle is to the triangle ABC.

IF two parallelograms be equiangular, as a side of the

first to a fide of the second, so is the other fide of the second to the straight line to which the other fide of the first has the same ratio which the first parallelogram has to the second. And consequently, if the ratio of the first parallelogram to the second be given, the ratio of the other side of the first to that straight line is given ; and if the ratio of the other side of the first to that straight line be given, the ratio of the first parallelogram to the second

is given.

Let AC, DF be two equiangular parallelograms, as BC, a fide of the first, is to EF, a Gde of the second, so is DE, the other Gde of the second, to the straight line to which AB, the o

a 14. 6.

ther side of the first has the same ratio which AC has to DF.

Produce the straight line AB, and make as BC to EF, fo
DE to BG, and complete the parallelo-
gram BGHC ; therefore, because BC, or

A
GH, is to EF, as DE to BG, the fides
about the equal angles BGH, DEF are
reciprocally proportional; wherefore • B
the parallelogram BH is equal to DF; G

11
and AB is to BG, as the parallelogram
AC is to BH, that is, to DF; as there-

D
fore BC is to EF, so is DE to BG, which
is the straight line to which AB has the E

F
fame ratio that AC has to DF.

And if the ratio of the parallelogram AC to DF be given, then the ratio of the straight line AB to BG is given; and if the ratio of AB to the straight line BG be given, the ratio of the parallelogram AC to DF is given.

74. 73.

PRO P. LXIV.

See N.

IF two parallelograms have unequal, but given angles,

and if as a fide of the first to a fide of the second, so the other side of the second be made to a certain straight line ; if the ratio of the first parallelogram to the second be given, the ratio of the other side of the first to that straight line shall be given. And if the ratio of the other fide of the first to that straight line be given, the ratio of the first parallelogram to the second shall be given.

Let ABCD, EFGH be two parallelograms which have the unequal, but given, angles ABC, EFG ; and as BC to FG, co make EF to the straight line M. If the ratio of the parallelo. gram AC to EG be given, the ratio of AB to M is given.

At the point B of the straight line BC make the angle CBK equal to the angle EFG, and complete the parallelogram KBCL. And because the ratio of AC to EG is given, and that AC is equal to the parallelogram KC, therefore the ratio of

KC to EG is given ; and KC, EG are equiangular ; thereb 63. dat, fore as BC to FG, so is b EF to the straight line to which KB

has a given ratio, viz. the same which the parallelogram KC has to EG: But as BC to FG, so is EF to the straight line M; therefore KB has a given ratio to M; and the ratio

of

£ 35. I.

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of AB to BK is given, because the triangle ABK is given in c 43. dat. species; therefore the ratio of AB to M is given.

d 9. dar. And if the ratio of AB to M be given, the ratio of the parallelogram AC to EG is given ; for since the ratio of KB to BA is given, as also the ratio of AB

K A L D to M, the ratio of KB to M is given d; and because the parallelograms KC, EG are cquiangular, as BC to FG, so is • EF to the straight line to which KB

B

b 63. dat

C has the same ratio which the parallelo- E

H gram KC has to EG; but as BC to FG, fo is EF to M; therefore KB is to M, M

G as the parallelogram KC is to EG; and the ratio of KB to M is given, therefore the ratio of the paral. lelogram KC, that is, of AC to EG, is given.

COR. And if two triangles ABC, EFG have two equal angles, or two unequal, but given, angles ABC, EFG, and if as BC a side of the first to FG a side of the second, so the other fide of the second EF be made to a straight line M; if the ratio of the triangles be given, the ratio of the other side of the first to the straight line M is given.

Complete the parallelograms ABCD, EFGH ; and because the ratio of the triangle ABC to the triangle EFG is given, the ratio of the parallelogram AC to EG is given, because the pa-c 15. 5. rallelograms are double f of the triangles ; and because BC is to f 41. J. FG, as EF to M, the ratio of AB to M is given by the 63d dat. if the angles ABC, EFG are equal ; but if they be unequal, but given angles, the ratio of AB to M is given by this propolition.

And if the ratio of AB to M be given, the ratio of the pa. rallelogram AC co EG is given by the same propositions; and therefore the ratio of the triangle ABC to EFG is given.

IF two cquiangular parallelograms have a given ratio to

one another, and if one side has to one fide a given ratio ; the other side shall also have to the other side a given ratio.

Let the two equiangular parallelograms AB, CD have a given ratio to one another, and let the fide EB have a given ratio to the Gde FD; the other side AE has also a given ratio to the other fide CF.

Dd

Because

Because the two equiangular parallelograms AB, CD have a

given ratio to one another; as EB, a side of the first, is to FD, a 63. dat.

a fide of the second, so is . FC, the other side of the second, to
the straight line to which AE, the other side of the first, has
the same given ratio which the first parallelogram AB bas
to the other CD. Let this straight line be EG ; therefore the
ratio of AE to EG is given ;
and EB is to FD, as FC to A

C
EG, therefore the ratio of
FC to EG is given, because

E

BF the ratio of EB to FD is gi

D ven; and because the ratio of G AE to EG, as also the ratio

of FC to EG is given ; the HKL b g. dat. ratio of AE to CF is given b.

The ratio of AE to CF may be found thus : Take a straight line H given in magnitude; and because the ratio of the parallelogram AB to CD is given, make the ratio of H to K the fame with it. And because the ratio of FD to EB is given, make the ratio of K to L the same : The ratio of AE to CF is the same with the ratio of H to L. Make as EB to FD, fo FC to EG, therefore, by inversion, as FD to EB, fo is EG to FC; and as AE to EG, fo is (the parallelogram AB to CD, and so is) H to K; but as EG to FC, so is (FD to EB, and so is) K to L ; therefore, ex aequali, an AE to FC, so is H to L.

IF two parallelograms have unequal, but given angles,

and a given ratio to one another ; if one side has to one side a given ratio, the other side has also a given ratio to the other side.

Let the two parallelograms ABCD, EFGH which have the given unequal angles ABC, EFG, have a given ratio to one asother, and let the ratio of BC to FG be given; the ratio also of AB to EF is given.

At the point B of the straight line BC make the angle CBK equal to the given angle EFG, and complete the parallelo

gram BKLC; and because each of the angles BAK, AKB is 2 43. dat. given, the triangle ABK is given in species; therefore the ratio of AB to BK is given ; and because, by the hypothesis,

the