c. 1. Dat. given c. but as LM to MN, fo is GH to HK; wherefore the ratio of GH to HK is given. See N. a. 22. I. b. 8. 1. 39. c. 1. Def. d. 1. Dat. e. 3. Def. 40. PROP. XLII. each of the fides of a triangle be given in magnitude; the triangle is given in species. Let each of the fides of the triangle ABC be given in magnitude; the triangle ABC is given in fpecies. a Make a triangle DEF the fides of which are equal, each to each, to the given straight lines AB, BC, CA; which can be done, because any two of them must be greater than the third ; and let DE be equal to AB, EF to BC, and FD to CA. and because the two fides ED, DF are equal to the two BA, AC, each to each, and the bafe EF equal to the base BC; the angle EDF is 44 equal to the angle BAC. therefore because the angle EDF, which is equal to the angle BAC, has been found, the angle BAC is given ©, in like manner the angles at B, C are given. and because the fides AB, BC, CA are given, their ratios to one another are given. therefore the triangle ABC is given in species. PROP. XLIII. F each of the angles of a triangle be given in magnitude; the triangle is given in species. Let each of the angles of the triangle ABC be given in magni tude; the triangle ABC is given in fpecies. Take a straight line DE given in pofition and magnitude, and at the a. 23. 1. points D, E make a the angle EDF equal to the angle BAC, and the angle A D other angles EFD, BCA are equal. B CE F and each of the angles at the points A, B, C is given, wherefore DEF equal to ABC; therefore the each of those at the points D, E, F is given. and because the straight line FD is drawn to the given point D in DE which is given in pofition, making the given angle EDF; therefore DF is given in pofition b. in like manner EF alfo is given in pofition; wherefore b. 32. Dat. the point F is given. and the points D, E are given; therefore each c of the straight lines DE, EF, FD is given in magnitude. where- c. 29. Dat. fore the triangle DEF is given in fpecies ; and it is fimilar to the d. 42. Dat. triangle ABC; which therefore is given in fpecies. I PROP. XLIV. F one of the angles of a triangle be given, and if the fides about it have a given ratio to one another; the triangle is given in fpecies. Let the triangle ABC have one of its angles BAC given, and let the fides BA, AC about it have a given ratio to one another; the triangle ABC is given in fpecies. A D CE F e. 41. 54.6. 1. Def. 6. 2. 32. Dat. Take a straight line DE given in pofition and magnitude, and at the point D in the given straight line DE made the angle EDF equal to the given angle BAC; wherefore the angle EDF is given. and because the straight line FD is drawn to the given point D in ED which is given in position, making the given angle EDF; therefore FD is given in pofition . and because the ratio of BA to AC is given, make the ratio of ED to DF the fame with it, and join EF. and because the ratio of ED to DF is given, and ED is given, therefore b DF is given in magnitude; and it is given alfo in position, and the point D is given, wherefore the point F is given . and the points D, E are given, c. 30. Dat. wherefore DE, EF, FD are given in magnitude; and the triangle d. 29. Dat. DEF is therefore given in fpecies. and because the triangles ABC, e. 42. Dat. DEF have one angle BAC equal to one angle EDF, and the fides about these angles proportionals; the triangles are f fimilar. but f. 6. 6. the triangle DEF is given in species, and therefore also the triangle ABC. B b. 2. Dat. See N. 42. IF PROP. XLV. the fides of a triangle have to one another given ratios; the triangle is given in fpecies. Let the fides of the triangle ABC have given ratios to one another. the triangle ABC is given in fpecies. Take a ftraight line D given in magnitude; and because the ratio of AB to BC is given, make the ratio of D to E the same B. 2. Dat. with it; and D is given, therefore a E is given. and because the ratio of BC to CA is given, to this make the ratio of E to F the fame; and E is given, and therefore a F. and because as AB to BC, fo is D to E, by compofition AB and BC together are to BC, as D and E to E; but as BC to CA, fo is E to F; therefore, ex C. 20. I. d. A. 5. b. 22. 5. aequali b, as AB and BC are to CA, fo are D and E to F. and AB and BC are greater than CA, therefore D and E are greater d than F. in the fame manner any two of the three D, E, F are greater than the third. make e. 22. 5. the B H A G DEF Дв K triangle GHK whofe fides are equal to D, E, F, fo that GH be. equal to D, HK to E, and KG to F. and because D, E, F are, each of them, given, therefore GH, HK, KG are each of them £42. Dat. given in magnitude; therefore the triangle GHK is given f in species. but as AB to BC, so is (D to E, that is) GH to HK; and as BC to CA, fo is (E to F, that is) HK to KG; therefore, ex aequali, as AB to AC, fo is GH to GK. wherefore the triangle ABC is equiangular and fimilar to the triangle GHK. and the triangle GHK is given in fpecies; therefore also the triangle ABC is given in species. g. 5. 6. COR. If a triangle is required to be made the fides of which fhall have the fame ratios which three given straight lines D, E, F have to one another; it is necessary that every two of them be greater than the third. PROP. XLVI. the fides of a right angled triangle about one of the acute angles have a given ratio to one another; the triangle is given in fpecies. Let the fides AB, BC about the acute angle ABC of the triangle ABC which has a right angle at A, have a given ratio to one another; the triangle ABC is given in fpecies. 43. Take a straight line DE given in pofition and magnitude; and because the ratio of AB to BC is given, make as AB to BC, fo DE to EF; and because DE has a given ratio to EF, and DE is given, therefore a EF is given. and because as AB to BC, DE to EF, and AB is lefs ↳ than BC, therefore DE is lefs EF. from the point D draw DG at right angles to DE, and from C. A. 5. the center E at the distance EF describe a circle which shall meet b DG in two points, let G be either of them, and join EG; therefore the circumference of the circle is given & in pofition. and the ftraight line DG is given in position, because it B fo is a. 2. Dat. than b. 19. 1. c. is drawn to the given point D in DE given in pofition, in a given angle. therefore f the point G is given. and the points D, E are f. 28. Dat. given, wherefore DE, EG, GD are given 8 in magnitude, and the g. 29. Dat. triangle DEG in fpecies h. and because the triangles ABC, DEG h. 42. Dat. have the angle BAC equal to the angle EDG, and the fides about the angles ABC, DEG proportionals, and each of the other angles BCA, EGD less than a right angle; the triangle ABC is equiangular and similar to the triangle DEG. but DEG is given i. 7. 6. in fpecies, therefore the triangle ABC is given in species. and in the fame manner, the triangle made by drawing a straight line from E to the other point in which the circle meets DG is given in fpecies. i Bb 4 See_N. 44. PROP. XLVII. F a triangle has one of its angles which is not a right angle given, and if the fides about another angle have a given ratio to one another; the triangle is given in fpecies. Let the triangle ABC have one of its angles ABC a given, but not a right angle, and let the fides BA, AC about another angle BAC have a given ratio to one another; the triangle ABC is given in fpecies. First, Let the given ratio be the ratio of equality, that is, let the fides BA, AC and confequently the angles ABC, ACB be equal. and because the angle ABC is given, the angle ACB, a. 32 I. and alfo the remaining a angle BAC is given. b. 43. Dat. therefore the triangle ABC is given b in fpecies. and it is evident that in this cafe the given angle ABC must be acute. A B C Next, Let the given ratio be the ratio of a less to a greater, that is, let the fide AB adjacent to the given angle be less than the fide AC. take a straight line DE given in pofition and magnitude, and make the angle DEF equal to the given angle ABC; therefore EF C c. 32. Dat is given in pofition. and because A B D the ratio of BA to AC is given, as d. 2. e. A. 5. f. 6. Def. because the circle is given in pofi tion, as alfo the ftraight line EF, the G g. 28. Dat. point F is given 8. and the points D, E are given, wherefore the h. 29. Dat. ftraight lines DE, EF, FD are given h in magnitude, and the tri1. 42. Dat. angle DEF in species i. and because BA is less than AC, the angle ACB is lefs than the angle ABC, and therefore ACB is lefs |