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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.

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PROPOSITION XII. THEOREM

277. Equimultiples of two quantities are in the same ratio as the quantities.

Hyp. a and b are two quantities.

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[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.

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Let m be a common measure contained in AD five times and in DB three times.

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Through the points of division of AB draw parallels to BC. These lines divide AE into five parts and EC into three parts,

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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

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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.

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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.

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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

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Given. Three lines m, n, and p.

Required. The fourth proportional to m, n, and p.

Construction. Draw any angle KAH.

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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

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