Ex. 528. If 2:3 =x and 2:274:y, find x. 1 y PROPOSITION XI. THEOREM 276. If four quantities are in proportion, like powers or like roots of these quantities are in proportion. PROPOSITION XII. THEOREM 277. Equimultiples of two quantities are in the same ratio as the quantities. Hyp. a and b are two quantities. [The proof is left to the student.] 278. DEF. If in a line AB, or its prolongation, a point C be taken, AC and BC are called segments of the line. 279. The segments are internal or external ones, according as C lies in AB or in the prolongation of AB. PROPORTIONAL LINES PROPOSITION XIII. THEOREM 280. A line parallel to one side of a triangle divides the other two sides proportionally. Let m be a common measure contained in AD five times and in DB three times. Through the points of division of AB draw parallels to BC. These lines divide AE into five parts and EC into three parts, By increasing the number of parts into which AD is divided, we can diminish the length of these parts, and therefore the length of B'B indefinitely. Hence DB' approaches DB as a limit, and EC' approaches EC as a limit. The variables AD AE and being always equals, must have DB' EC 281. SCHOLIUM. If the transversal intersects two sides of the triangle, these sides are divided internally, if it meets their prolongations, the sides are divided externally in the same ratio. 282. COR. If a line parallel to one side of a triangle intersects the other two sides, either side is to one of its segments as the other side is to its corresponding segment. Ex. 531. In the diagram for Prop. XIII, if AD = 4, DB = 8, AE 3, find EC. = Ex. 532. In the same diagram, find DB, if AD = a, AE = b, and EC = c. Ex. 533. In the same diagram, find AE, if AB AC 9. = 12, AD = 8, and Ex. 534. In the same diagram, find EC, if AB=m, AD = n, and AC = p. Ex. 535. In the same diagram, find AD, if AD AE= 9. Ex. 536. In the same diagram, find AE, if AE EC = 20. = EC, DB = 4, and = 2 DB, AD = 10, and Ex. 537. Three or more parallels make proportional segments on the sides of an angle. Ex. 538. In the diagram for Prop. XIII, Case II, find B'B if AD = 3, AE 4, and C'C = 1. = Ex. 539. In the same diagram find EC' if AD and C'C= 1. Ex. 540. Three or more parallels make proportional segments on any two transversals. Ex. 541. If in the diagram for Prop. XIII, AD = 2, DB = 3, AE = 4, and EC 4, is DE parallel to BC? = PROPOSITION XIV. PROBLEM 283. To find the fourth proportional to three given Given. Three lines m, n, and p. Required. The fourth proportional to m, n, and p. Construction. Draw any angle KAH. Through C, draw a line parallel to BD, meeting AH in E. DE is the required fourth proportional. [The proof is left to the student.] Ex. 542. Find the third proportional to two given lines. Ex. 543. If a, b, and c are given lines, construct a line x, so that a: b = x:C. bc Ex. 544. If a, b, and c are given lines, construct a line equal to a |