Proposition 7. 289. If four quantities are in proportion, they are in proportion by composition and division; that is, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference. 290. The products of the corresponding terms of two or more proportions are proportional. 291. A greater quantity is said to be a multiple of a less, when the greater contains the less an exact number of times. Equimultiples of two quantities are quantities which contain them the same number of times. Thus, ma and mb are equimultiples of a and b. Proposition 9. 292. When four quantities are in proportion, if the first and second be multiplied, or divided, by any quantity, as also the third and fourth, the resulting quantities will be in proportion. Multiply both terms of the first fraction by m, and both terms of the second by n. 293. SCH. Either m or n may be unity. In a similar manner it may be shown that if the first and third terms be multiplied, or divided, by any quantity, and also the second and fourth, the resulting quantities will be in proportion. That is, equimultiples of two quantities are in the same ratio as the quantities themselves. Proposition 10. 295. If four quantities are in proportion, their like powers, or roots, are in proportion. Hyp. Let To prove and Proof. an: bn = cn: dn. a b Raising to the nth power, an bn Extracting the nth root, Proposition 11. 296. If any number of quantities are in proportion, any antecedent is to its consequent, as the sum of any number of the antecedents is to the sum of the corresponding consequents. Hyp. To prove Let a b c d e f etc. abcd etc. = a+c+e+ etc.: b+d+f+ etc. adbe, and af= be, etc., etc. Adding, a(b+d+f+ etc.) = b(a + c + e + etc.) = (281) ..a b c d etc.=a+c+e+etc.: b+d+f+etc. (283) PROPORTIONAL LINES. Q.E. D. 297. DEF. Two straight lines are said to be cut proportionally when the parts of one line are in the same ratio as the corresponding parts of the other line. Thus, AB and CD are cut pro portionally at P and Q if AP: PB CQ : QD. P A B Proposition 12. Theorem. 298. A straight line parallel to one side of a triangle · divides the other two sides proportionally. Hyp. Let DE be || to BC in the ▲ ABC. Το prove AD: DB = AE: EC. CASE I. When AD and DB are commensurable. Proof. Take AH, any common measure of AD and DB, and suppose it to be contained 4 times in AD and 3 times in DB. B H E Then AD 4 = 3 (1) Through the several pts. of division of AD, DB draw || s to BC. They will divide AE into 4 equal parts and EC into 3 equal parts. If || s intercept equal lengths on any transversal, they intercept equal lengths on every transversal (152). ... AD : DB = AE : EC. CASE II. When AD and DB are incommensurable. Proof. In this case we know (232) that we may always find a line AG as nearly equal as we please to AD, and such that AG and GB are commensurable. Draw GH || to BC; then Now, these two ratios being always equal while the common measure is indefinitely diminished, they will be equal when G moves up to and as nearly as we please coincides with D. 299. COR. 1. By composition (287), we have AD+ DB: AD AE+ EC: AE, = AB: AD = AC: AE. (233) Q.E.D or Also, (288) (286) AB, CD are cut by any number of parallels, AC, EF, GH, BD, the corresponding intercepts are proportional. For, let AB and CD meet at O. Then, by (299), Similarly, ... AE : CF = EG : FH. EG FH GB: HD. EXERCISES. 1. From a point E in the common base AB of two triangles ACB, ADB, straight lines are drawn parallel to AC, AD, meeting BC, BD at F, G: show that FG is parallel to CD. 2. In a triangle ABC the straight line DEF meets the sides BC, CA, AB at the points D, E, F respectively, and it makes equal angles with AB and AC: prove that BD: CD BF: CE. |