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46.

PROP.

XLIX.

IF
F a triangle has one angle given, and if the sides a-

bout another angle, both together, have a given 1atio to the third side; the triangle is given in fpecies.

B

Let the triangle ABC have one angle ABC given, and let the two sides BA, AC about another angle BAC have a given ratio to BC; the triangle ABC is given in species.

Suppose the angle BAC to be bifected by the straight line AD; BA and AC together are to BC, as AB to BD, as was thewo in the preceding Proposition. but the ratio of BA and AC together

to BC is given, therefore also the ratio of AB to BD is given. and 2.44. Dat. the angle ABD is given, wherefore the triangle ABD is given in

species; and consequently the angle BAD,
and its double the angle BAC are given;

A
and the angle ABC is given. therefore
b. 43. Dat the triangle ABC is given in species b.

A triangle which shall have the things mentioned in the Proposition to be given,

C may be thus found. Let EFG be the gi

E

H ven angle, and the ratio of H to K the

K given ratio ; and by Prop. 44. find the triangle EFL which has the angle EFG for one of its angles, and the ratio of the F L G sides EF, FL about this angle the same with the ratio of H to K; and make the angle LEM equal to the angle FEL. and because the ratio of H to K is the ratio wbich two fides of a triangle have to the third, H must be greater than K; and because EF is to FL, as H to K, therefore EF is greater than FL, and the angle FEL, that is LEM is therefore lets than the angle ELF. wherefore the angles LFE, FEM are less than two right angles, as was shewn in the foregoing Proposition, and the straight lines FL, EM muft meet if produced ; let them meet in G. EFG is the triangle which was to be found; for EFG is one of its angles, and because the angle FEG is bifected by EL, the two fides TE, EG together have to the third fide FG the ratio of EF to Fl., that is the given ratio of H to K.

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IF from the vertex of a triangle given in species, a

straight line be drawn to the base in a given angle; it shall have a given ratio to the base.

From the vertex A of the triangle ABC which is given in fpecies, let AD be drawn to the base BC in a given angle ADB; the ratio of AD to BC is given. Because the triangle ABC is given in

A fpecies, the angle ABD is given, and the angle ADB is given; therefore the triangle ABD is given in species; wherefore the

a. 43. Dat. ratio of AD to AB is given. and the ratio B D C of AB to BC is given ; and therefore b the ratio of AD to BC b.9. is given.

Dat.

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RECTILINE A L, figures given in fpecies, are di

vided into triangles which are given in species.

Let the rectilineal figure ABCDE be given in species; ABCDE may be divided into triangles given in species.

Join BE, BD, and because ABCDE is given in species, the angle BAE is given', and the ratio of BA

d. 3. Def,

Α. to BE is given *; wherefore the triangle BAE is given in species, and the angle

b. 44. Dat. AEB is therefore given", but the whole

B angle AED is given, and therefore the

E remaining angle BED is given. and the ratio of AE to EB is given, as also the

C D ratio of AE to ED; therefore the ratio of BE to ED is given. c. 9. Dato and the angle BED is given, wherefore the triangle BED is given bin fpecies. in the fame manner the triangle BDC is given in species. therefore rectilineal figures which are given in species are divided into triangles given in species.

48.

I

PRO P. LII.
F two triangles given in fpecies be described

upoг. the fane firaight line ; they shall liave a given ratio to one another.

Let the triangles ABC, ABD given in species be difcribed upoz the fame straight line AB; the ratio of the triangle ABC to the triangle ABD is given.

Thro’ the point C draw CE parallei to AB, and let it meet DA produced in E, and join BE. because the triangle ABC is given in fpecies, the angle BAC, that is the angle ACE, is given ; and because the triangle ABD is given in species, the angle DAB, that is the angle AEC is given. therefore the tri

I

1 angle ACE is given in Species; wherefore the

ratio of EA tO AC is A a. 3. Def. given", and the ratio

of CA to AB is given,

as alio the ratio of BA b.9. Dat. to AD; therefore b the ratio of EA to AD is given. and the tri

angle ACB is equal to the triangle ALB, and as the triangle AEB, or ACB, is to the triangle ADB, fo is d the straight lice LA to AD. but the ratio of EA to AD is given, therefore the ratio of the triangle ACB to the triangle ADB is given.

PROBLEM. To find the ratio of two triangles ABC, ABD given in species, and which are described upon the same straight line AB.

Take a ftraight line FG given in position and magnitude, and because the angles of the triangles ABC, ABD are given, at the points F, G of the straight line FG make the angles GFH, GFK equal to the angles BAC, BAD; and the angles FGH, FGK equal to the angles ABC, ABD, each to each. therefore the triangles ABC, ABD are cquiangular to the triangles FGH, TGK, each to each. through the point II draw in parallel to FG meeting KF produced in L. and because the angles DAC, BAD are equal to the angles GFH, GTK, each to cach; therefore the angles ACE, AEC are equal to FHL, FLH, each to each, and the triangle AEC equiangular to the triangle FLH. therefore as EA :0 AC,

e. 37. I. d. 1. 6.

e. 22. 1.

fo is Lr to TH; and as CA to AB, fo HF to FG; and as BA to AD, fo GF to FK; wherefore, ex aequali, as EA to AD, fo is LF to FK. but, as was fhewn, the triangle ABC is to the triangle ABD, as the straight line EA to AD, that is as LF to FK. the ratio therefore of LF tɔ FK has been found which is the fame with the ratio of the triangle ABC to the triangle ABD.

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C. 7. Dat.

PROP. LIII.

49. T two rectilineal figures given in species be described Sce N.

uron the fame straight line; they fall have a given ratio to one another.

Let any two rectilineal figures ABCDE, ADFG which are civen in species, be described upon the fame straight line AB; the ratio of them to one another is given.

Join AC, AD, AF; each of the triangles AED, ADC, ACB, AGF, ABF is given ? in species. and because the triangles ADE, a. 51. Dat. ADC given in fpecies are described upon the fame straight line AD, the ratio of EAD to DAC is given b; and,

b. 52. Dat. by compofition, the ratio of LACD to

c
DAC is given. and the ratio of DAC
to CAB is given", because they are de-

A
fcribed upon the same straight line
AC; therefore the ratio of EACD to
ACE is given d; and, by composition, H.

SMO ... Dat

d. ,. Dat. the ratio of ABCDE to ABC is given. in the fame inanner, the ratio of ABFG to ABF is given. but the ratio of the triangle ABC to the triangle ABF is given b; vhciefore because the ratio of ABCDE to ABC is given, as also the ratio of ABC to ABF, and the ratio of ABF to ABIG; tle ratio of the rectilineal AECDE to the rectilineal ALFG is given d.

PROBLEM. To find the ratio of two rectilinc::l figures given in speciao, and defcribed upon the same straight line.

Lct ARCDE, ABFG be two rectilincal figures given in fpecie, and described upon the fame straight line AB, and join AC, AD, AF. take a straight line TIK given in position and magnitude, and by the 51. Dat find the ratio of the triangle ADE to the unele ADC, and make the ratio of HK to KL the fame with it. fidanti

the ratio of the triangle ACD to the triangle ACB, and make the
ratio of KL to LM the same. also, find the ratio of the triangle
ADC to the triangle ABF, and inake the ratio of LM to MN the
fame. and lally, find the ratio of the triangle AFB to the triangle
AFG, and make the ratio of MN to
NO the same, then the ratio of ABCDE
to A BFG is the fime with the ratio
of HM to MO.

Because the triangle EAD is to the
triangle DAC, as the straight line HK А
to KL; and as triangle DAC to CAB,

G

F fo is the straight line KL to LM;

KLMN therefore by using composition as of. H ten as the number of triangles requires, the rectilineal ABCDE is to the triangle ABC, as the straight line HM to ML. in like manner, because triangle GAF is to FAB, 25 ON to NM, by composition, the rectilineal ABFG is to the triangle ABF, as MO to MN; and, by inversion, as ABF to ABFG, fo is NM to MO. and the triangle ABC is to ABF, as LM to MN. wherefore because as ABCDE to ABC, so is HM to ML; and as ABC to ABF, fo is LM to MN; and as ABF to ABFG, so is MN to MO; ex aequali, as the rectilineal ABCDE to ABFG, so is the straight line HM to MO.

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IF

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PRO P. LIV. 50.

F two straight lines have a given ratio to one ano

ther; the similar rectilineal figures described upon them fimilarly, shall have a given ratio to one another.

Let the straight lines AB, CD have a given ratio to one another, and let the fimilar and similarly placed rectilineal figures E, F be described upon them; the ratio of E to F is given.

TO AB, CD let G be a third pro-
portional; therefore as AB to CD, so
is CD to G, and the ratio of AB to

JE
CD is given, wherefore the ratio of

A
CD to G is given ; and consequently

H K a. g. Dat. the ratio of AB to G is also given'.

but as AB to G, so is the figure E to b. 2. Cor. the figure 6 F. therefore the ratio of E to F is given.

F

20. 6,

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