BG, as KB to BC; as therefore KB to BC, fo is LM to Q: and because BE is to BK, as LO to LM; and as BK to BC, fo is LM to Q; and as BC to BA, fo Q was made to R ; therefore, ex æquali, as BE to BA, fo is LO to R, that is to LN; and the angles ABE, NLO are equal; therefore the parallelogram BF is fimilar to LS. PROP. LXX. 62. 78. IF two straight lines have a given ratio to one another, see N. and upon one of them be defcribed a rectilineal figure given in species, and upon the other a parallelogrami having a given angle; if the figure have a given ratio to the parallelogram, the parallelogram is given in species. Let the two ftraight lines AB, CD have a given ratio to one another, and upon AB let the figure AEB given in fpecies 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 fpecies. Upon the ftraight line AB, conceive the parallelogram AG to be defcribed fimilar, and fimilarly placed to FD; and because the ratio of AB to CD is given, and upon them are defcribed the fimilar rectilineal figures AG, FD; the ratio of AG to FD is gi ven a; and the ratio of FD to AEB A is given; therefore b the ratio of AEB to AG is given; and the angle ABG is given, because it is equal to the angle FCD; becaufe therefore the parallelogram AG which has a given angle ABG is applied to a fide AB of the figure AEB given in fpe M E F B a 54. dat. b 9. dat. GC D N HKL cies, and the ratio of AEB to AG is given, the parallelogram AG is given in fpecies; but FD is fimilar to AĞ; therefore e 69. data FD is given in fpecies. A parallelogram fimilar to FD may be found thus; take a ftraight 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 fame 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 defcribed upon CD has to the figure AG defcribed upon AB fimilar to FD; and make the ratio of K to L the fame with this ratio: And because the ratios of H to K, and of K Dd4 b 9. dat. to L are given, the ratio of H to L is given b; because there. fore, as AEB to FD, fo is H to K; and as FD to AG, fo is K to L; ex æquali, as AEB to AG, fo is H to L; therefore the ratio of AEB to AG is given; and the figure AEB is given in fpecies, and to its fide AB the parallelogram AG is applied in the given angle ABG; therefore by the 69th dat. a parallelogram may be found fimilar to AG: Let this be the parallelogram MN; MN alfo is fimilar to FD; for, by the conftruction, MN is fimilar to AG, and AG is fimilar to FD; therefore the parallelogram FD is fimilar to MN. PROP. LXXI. IF the extremes of three proportional straight lines have given ratios to the extremes of other three proportional ftraight lines; the means fhall also have a given ratio to one another: And if one extreme has a given ratio to one extreme, and the mean to the mean; likewife the other extreme fhall have to the other a given ratio. Let A, B, C be three proportional ftraight 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 alfo of C to F is given, the a 67. dat. ratio of the rectangle A, C to the rectangle D, F is given 2; b. 17. 6. but the fquare of B is equal b to the rectangle A, C; and the fquare of E to the rectangle b D, F; therefore the ratio of the 58. dat. fquare of B to the fquare of E is given; wherefore e alfo the ratio of the ftraight line B to E is given. Next, let the ratio of A to D, and of B to E be gi- 54. dat. the fquare of B to the fquare of E is given d; there AB C 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 hall have a given ratio to the other mean, as may be fhewn in the fame manner as in the foregoing propofition. PROP. PROP. LXXII. JF four straight lines be proportionals; as the first is to the straight line to which the fecond has a given ratio, fo is the third to a straight line to which the fourth has a given ratio. Let A, B, C, D be four proportional ftraight lines, viz. as A to B, fo C to D; as A is to the ftraight line to which B has a given ratio, so is C to a feraight line to which D has a given ratio. Let E be the ftraight line to which B has a given. ratio, and as B to E, fo make D to F: The ratio of B to E is given 2, and therefore the ratio of D to F; and because as A to B, fo is C to D; and as B to E. fo D to F; therefore, ex æquali, as A to E, fo is ABE C to F; and E is the ftraight line to which B has a CDF IF PROP. LXXIII. Y. F four straight lines be proportionals; as the firt is to See N. the straight line to which the fecond has a given ratio, fo is a ftraight line to which the third has a given ratio to the fourth. Let the ftraight line A be to B, as C to D; as A to the ftraight line to which B has a given ratio, fo 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, fo 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 BABE to E, fo F to C; therefore, ex æquali in proportione FC D perturbata a, A is to E, as F to D; that is, Á is to L to which B has a given ratio, as F, to which C has a given ratio, is to D. a 23. Si PROP. 64 2 12. 2. CI. 6. IF PROP. LXXIV. F a triangle has a given obtufe angle; the excess of the fquare of the fide which fubtends the obtufe angle, above the fquares of the fides which contain it, fhall have a given ratio to the triangle. Let the triangle ABC have a given obtufe angle ABC; and produce the straight line CB, and from the point A draw AD perpendicular to BC: The excefs of the fquare of AC above the fquares of AB, BC, that is a, the double of the rectangle contained by DB, BC, has a given ratio to the triangle ABC. Because the angle ABC is given, the angle ABD is alfo given and the angle ADB is given; wherefore the triangle 43. dat. ABD is given b in fpecies; and therefore the ratio of AD to DB is given: And as AD to DB, fo is the rectangle AD, BC to the rectangle DB, BC; wherefore the ratio of the rectangle AD, BC to the rectangle DB, BC is given, as alfo the ratio of twice the rectangle DB, BC to the rectangle AD, BC: But the ratio of the rectangle AD, BC to the triangle ABC is given, because it is double d of the triangle; therefore the ratio of twice the rectangle DB, BC 41.1. e 9. dat. to the triangle ABC is given ; and twice f 6. 4 E H FG C the rectangle DB, BC is the excefs a of D B And the ratio of this excefs to the triangle ABC may be found thus; take a ftraight line EF given in pofition and mag. nitude; and because the angle ABC is given, at the point F of the ftraight line EF, make the angle EFG equal to the angle ABC; produce GF, and draw EH perpendicular to FG; then the ratio of the excefs of the fquare of AC above the squares of AB, BC to the triangle ABC is the fame with the ratio of quadruple the ftraight line HF to HE. Because the angle ABD is equal to the angle EFH, and the angle ADB to EHF, each being a right angle; the triangle ADB is equiangular to EHF; therefore f as BD to DA, g Cor. 4. 5. fo FH to HE; and as quadruple of BD to DA, fo is g quadruple of FH to HE: But as twice BD is to DA, fo is c twice the rectangle DB, BC to the rectangle AD, BC; and as DA to the half of it, fo is h the rectangle AD, BC to its half the triangle triangle ABC; therefore, ex aequali, as twice BD is to the half of DA, that is, as quadruple of BD is to DA, that is, as quadruple of FH to HE, fo is twice the rectangle DB, BC to the triangle ABC. PROP. LXXV. IF a triangle has a given acute angle; the space by which the fquare of the fide fubtending the acute angle is lefs than the fquares of the fides which contain it, fhall have a given ratio to the triangle. 65. Let the triangle ABC have a given acute angle ABC, and draw AD perpendicular to BC; the space by which the fquare of AC is lefs than the squares of AB. BC, that is a, the double 13. 4. of the rectangle contained by CB, BD, has a given ratio to the triangle ABC. A Because the angles ABD, ADB are each of them given, the triangle ABD is given in fpecies; and therefore the ratio of BD to DA is given: And as BD to DA, fo is the rectangle CB, BD to the rectangle CB, AD; therefore the ratio of thefe rectangles is given, as alfo the ratio of twice the rectangle CB, BD to the rectangle CB, AD; but the rectangle CB, AD has a given ratio to its half the triangle ABC; therefore b the ratio of twice the rectangle CB, BD to the triangle ABC is given; and twice the rectangle CB, BD is a the fpace by which the fquare of AC is lefs than the fquares of AB, BC; the efore the ratio of this space to the triangle ABC is given: And the ratio may be found as in the preceding propofition. IF LE M M A. B DC b 9. dat. from the vertex A of an ifofceles triangle ABC, any ftraight line AD be drawn to the bafe BC; the fquare of the file AB is equal to the rectangle BD, DC of the fegments of the bafe together with the fquare of AD; but if AD be drawn to the bafe produced, the fquare of AD is equal to the rectangle BD, DC together with the fquare of AB. A CAS. 1. Bifect the bafe BC in E, and join AE which will be perpendicular a to BC; wherefore the fquare of AB is equal 6 to the fquares of AE, EB; but the fquare of EB is equal c to the rectangle BD, DC together with the fquare of DE; therefore the fquare of AB is equal to the D BDE C fquares a &.. E b 4. 7. r. C S. r. |