Proposition X

282. 1. Form a proportion; multiply or divide the terms of either ratio by any number. How do the resulting ratios compare?

2. Transform similarly and investigate other proportions.

Theorem. If in a proportion the terms of either coup-let are multiplied by any quantity, the resulting ratios form a proportion.

Multiplying both terms of the first fraction by m,

283. Cor. If in a proportion the terms of either couplet are divided by any quantity, the resulting ratios form a proportion.

Proposition XI

284. 1. Form a proportion; multiply or divide the antecedents or the consequents by any number. How do the resulting ratios compare?

Theorem. If in any proportion the antecedents or the consequents are multiplied by the same quantity, the resulting ratios are in proportion.

285. Cor. If in any proportion the antecedents or the consequents are divided by the same quantity, the resulting ratios are in proportion.

Proposition XII

286. 1. Form several proportions. Multiply together their corresponding terms, and discover whether the resulting quantities form a proportion.

2. If there is an equal antecedent and consequent in the same couplet, or in corresponding couplets, cancel them from the products of the corresponding terms. Do the resulting quantities form a proportion?

Theorem. The products of the corresponding terms of any number of proportions are in proportion.

287. Cor. In finding the proportion formed by the products of the corresponding terms of any number of proportions, an equal antecedent and consequent in the same couplet, or in corresponding couplets, may be dropped.

Dividing the terms of the first couplet by b and the terms of the second by c, § 283,

a:e=f: d.

Proposition XIII

288. 1. Form a proportion; raise the terms of both ratios to the same power. How do the resulting ratios compare?

2. Extract the same root of the terms of both ratios in a proportion, as 4:9 16:36. How do the resulting ratios compare?

3. Transform similarly and investigate other proportions.

Theorem. In any proportion, like powers or like roots of the terms are in proportion.

Raising both fractions in (1) to the nth power,

Extracting the nth root of both fractions in (1),

(1)

Q.E.D.

Ex. 408. Make the changes that may be made upon the following proportion without destroying the equality of the ratios: 16:36

Ex. 409. If a: b = c: d, prove that ma: nb = mc : nd.

4:9.

= cd, prove that a + 4b: b c+ 4 d: a. bb : c, prove that a2 + ab : b2 + bc

Ex. 410. If a

b

Ex. 411. If a

b

Ex. 412. If a

:

Ex. 413. If a:

=

bc, prove that a: c = (a + b)2: (b + c)2.

:b =mn, and b c = no, prove that a: c = m : 0.

Ex. 415. If a b c d, prove that

3a+4b: 4 a − 5 b = 3 c +4d:4c-5 d

BOOK IV

PROPORTIONAL LINES AND SIMILAR FIGURES

Proposition I

289. 1. Draw a line parallel to the base of a triangle through the middle point of one side and cutting the other side. How do the segments of the other side compare in length?

2. Draw a line parallel to the base one fourth, one sixth, or any part of the distance from the extremity of the base to the vertex. How do the segments of the other side compare?

3. How does the ratio of one of these sides to either of its segments compare with the ratio of the other to its corresponding segment?

Theorem. A line which is parallel to one side of a triangle and meets the other two sides divides those sides proportionally.

Data: Any triangle, as ABC, and any line parallel to AB, as DE, meeting AC and BC in D and E, respectively.

D

E

To prove CD: DA=CE: EB.

A

M

B

Proof. Case I. When CD and DA are commensurable. Suppose that M is a unit of measure common to CD and DA, that M is contained in CD 3 times and in DA 2 times.

and

Divide CD and DA into parts each equal to the common measure M, and from each point of division draw lines parallel to AB. § 157, these lines divide CE into 3 and EB into 2 equal parts;

and, Ax. 1,

CE: EB

3:2,

CD: DACE: EB.

Case II. When CD and DA are incommensurable. Since CD and DA are incommensurable, suppose that CD and DF are commensur

Q

If M is indefinitely diminished, the ratios CD: DF and CE: EG remain equal, and indefinitely approach their limiting ratios CD: DA and CE : EB, respectively.

Hence, § 222,

CD: DA=CE: EB.

Therefore, etc.

Q.E.D.

290. Cor. A line which is parallel to one side of a triangle and meets the other two sides divides them so that one side is to either of its segments as the other side is to its corresponding segment.

Proposition II

291. Draw a line dividing each of two sides of a triangle into halves, or into other proportional parts. What is the direction of this line with reference to the third side?

Theorem. A line which divides two sides of a triangle proportionally is parallel to the third side. (Converse of Prop. I.)

Proof. If DE is not parallel to AB, some other line drawn

through D will be parallel to AB.

Suppose that DF is that line.