46. IF PRO P. XLIX. Fa triangle has one angle given, and if the fides about another angle, both together, have a given ratio to the third fide; the triangle is given in fpecies. Let the triangle ABC have one angle ABC given, and let the two fides BA, AC about another angle BAC have a given ratio to BC; the triangle ABC is given in fpecies. Suppofe the angle BAC to be bifected by the ftraight line AD; BA and AC together are to BC, as AB to BD, as was shewn in the preceeding Propofition. but the ratio of BA and AC together to BC is given, therefore alfo the ratio of AB to BD is given. and the angle a. 44. Dat. ABD is given, wherefore the triangle ABD is given in species; and confequently the angle BAD, and its double the angle BAC are given; and the angle ABC is given. therefore the b. 43. Dat. triangle ABC is given in fpecies b. B H K F A D C E M L G A triangle which fhall have the things mentioned in the Propofition to be given, may be thus found. Let EFG be the given angle, and the ratio of H to K the 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 fides EF, FL about this angle the fame 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 which 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 lefs than the angle ELF. wherefore the angles LFE, FEM are lefs than two right angles, as was fhewn in the foregoing Propofition, and the ftraight lines FL, EM must 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 FE, EG together have to the third fide FG the ratio of EF to FL, that is the given ratio of H to K. PROP. IF PRO P. L. from the vertex of a triangle given in fpecies, a ftraight line be drawn to the bafe 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 species, 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 fpecies, the angle ABD is given, and the angle ADB is given; therefore the triangle ABD is given in fpecies; wherefore the ratio of AD to AB is given. and the ratio of AB to BC is given; and therefore the ratio of AD to BC is given. B A 76. a. 43. Dat. D Cb. 9. Dat. PROP. LI. RECTILINEAL figures given in species, are divided in to triangles which are given in fpecies. Let the rectilineal figure ABCDE be given in fpecies; ABCDE may be divided into triangles given in fpecies. 47. Join BE, BD, and because ABCDE is given in species, the angle to ED; therefore the ratio of BE to ED is given . and the angle c. 9. Dat. BED is given, wherefore the triangle BED is given in fpecies. in the fame manner the triangle BDC is given in fpecies. therefore rectilineal figures which are given in fpecies are divided into triangles given in fpecies. PROP. 1 F two triangles given in fpecies be defcribed upon the fame ftraight line; they fhall have a given ratio to one another. Let the triangles ABC, ABD given in fpecies be described upon the fame ftraight line AB; the ratio of the triangle ABC to the triangle ABD is given. Thro' the point C draw CE parallel 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 becaufe the triangle ABD is given in species, the angle DAB, E that is the angle AEC is given. c. 37. 1. ratio of BA to AD; b. 9. Dat. therefore b the ratio of EA to AD is given. and the triangle ACB is equal to the triangle AEB, and as the triangle AEB, or ACB, is to the triangle ADB, fo is "the ftraight line EA to AD. but the ratio of EA to AD is given, therefore the ratio of the triangle ACB to the triangle ADB is given. d. 1. 6. PROBLEM. To find the ratio of two triangles ABC, ABD given in fpecies, and which are described upon the fame straight line AB. Take a ftraight line FG given in pofition and magnitude, and because the angles of the triangles ABC, ABD are given, at the points c. 23. 1. 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 equiangular to the triangles FGH, FGK, each to each. thro' the point H draw HL parallel to FG meeting KF produced in L. and because the angles BAC, BAD are equal to the angles GFH, GFK, each to each; 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 to AC, fo is LF to FH; and as CA to 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 ftraight line EA to AD, that is as LF to FK. the ratio therefore of LF to FK has been found which is the fame with the ratio of the triangle ABC to the triangle ABD. IF F two rectilineal figures given in fpecies be defcribed See N. upon the fame ftraight line; they shall have a given ratio to one another. Let any two rectilineal figures ABCDE, ABFG which are given in fpecies, be defcribed upon the fame ftraight line AB; the ratio of them to one another is given. Join AC, AD, AF; each of the triangles AED, ADC, ACB, a AGF, ABF is given in fpecies. and because the triangles ADE, a. 51. Dat. ADC given in fpecies are defcribed D E b. 52. Dat. C c. 7. Dat. A B F KL MN 9. Dat. G upon the fame ftraight line AD, the H+ PROBLEM. To find the ratio of two rectilineal figures given in fpecies, and defcribed upon the fame ftraight line. Let ABCDE, ARFG be two rectilineal figures given in fpecies, and defcribed upon the fame ftraight line AB, and join AC, AD, AF. take a straight line HK given in pofition and magnitude, and by the 52. Dat. find the ratio of the triangle ADE to the triangle ADC, and make the ratio of HK to KL the fame with it. find alfo the 2. 50. the ratio of the triangle ACD to the triangle ACB, and make the ra D E A B F G H KL MN O Because the triangle EAD is to the triangle DAC, as the ftraight line HK to KL; and as triangle DAC to CAB, fo is the straight line KL to LM; therefore by using compofition as often 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, as ON to NM, by composition, the rectilineal ABFG is to the triangle ABF, as MO to MN; and, by inverfion, 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, fo is HM to ML; and as ABC to ABF, fo is LM to MN; and as ABF to ABFG, fo is MN to MO; ex aequali, as the rectilineal ABCDE to ABFG, fo is the ftraight line HM to MO. IF PROP. LIV. F two ftraight lines have a given ratio to one another; the fimilar rectilineal figures defcribed upon them fimilarly, shall have a given ratio to one another. Let the ftraight lines AB, CD have a given ratio to one another, and let the fimilar and fimilarly placed rectilineal figures E, F be defcribed upon them; the ratio of E to F is given. To AB, CD let G be a third proportional; therefore as AB to CD, fo is CD to G. and the ratio of AB to CD is given, wherefore the ratio of AA CD to G is given; and confequently A Dat. the ratio of AB to G is also given. but as AB to G, fo is the figure E to B C D b. 2. Cor. the figure b F. therefore the ratio of E to F is given. 20.6. PRO |