Ex. 523. If x + y : y = 7:3, find the ratio of x and y.

Ex. 524. If x

y: y = 2:3, find the ratio of x and y.

PROPOSITION VIII. THEOREM

272. If four quantities are in proportion, they are in proportion by composition and division, i.e. the sum of the first two terms is to their difference as the sum of the last two terms is to their difference.

Ex. 526. If x + y : x − y = a:b, find the ratio of x to y.

Q.E.D.

PROPOSITION IX. THEOREM

273. In a continued proportion the sum of any number of antecedents is to the sum of the corresponding consequents as any antecedent is to its consequent.

274. The products of the corresponding terms of two or more proportions are in proportion.

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.

Proof. CASE I. AD and DB are commensurable.

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, all being equal.

Whence

(144)

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.