Plane and solid geometry . nally at D if this point is on the prolonga-tion of the line. The segments are AD and DB, It should be noted that in either case the point of division isone end of each segment. 407. Def. Two straight lines are divided proportionally if the ratio of one line to either of its segments is equal to theratio of the other line to its corresponding segment. 408. In Prop. XI, II, the following theorems (Appendix.§§ 586 and 591) will be assumed: (a) TJie quotient of a variable by a constant is a variable. (b) TJie limit of the quotient of a variable by a constant is thelimit
Plane and solid geometry . nally at D if this point is on the prolonga-tion of the line. The segments are AD and DB, It should be noted that in either case the point of division isone end of each segment. 407. Def. Two straight lines are divided proportionally if the ratio of one line to either of its segments is equal to theratio of the other line to its corresponding segment. 408. In Prop. XI, II, the following theorems (Appendix.§§ 586 and 591) will be assumed: (a) TJie quotient of a variable by a constant is a variable. (b) TJie limit of the quotient of a variable by a constant is thelimit of the variable divided by the constant Thus, if a; is a variable and k a constant: (1) - is a (2) If the limit of x is y, then the limit of - is |. Ex. 645. In the fit^re of § 409, name the segments into which ABIs divided by D; the segments into which AD is divided by B. BOOK m 171 Proposition- XL Theorem 409. A straight line parallel to one side of a triangledivides the other two sides Fig. 1. Given A ABC with line DE II BC. To prove AB __ACAD ~ AE L If AB and AD are commensurable (Fig. 1). Argument 1. Let AF be a common measure oi AB and AD, and suppose that AF is con-tained in AB r times and in AD stimes. 2. Theni^=^. AD S 3. Through the several points of division on AB, as F, G, etc., draw lines || BC. 4. These lines are || DE and to each other. 5. /. AC is divided into r equal parts and AE into s equal __rAE ~ sAB ACAD Ae 6. 7 . ^^^_? • • • ^~~^ — Reasons1. § 335. 2. §341. 3. § 179. 4. § 180. 5. § 244. 6. § 341, 7. § 54,1. 172 PLANE GEOMETRY II. If AB and AD are incommensurable (Fig. 2).
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