BOOK IV. SIMILAR FIGURES.

PROPORTIONS

SECTION I. RATIO AND PROPORTION

258. Terms of a Proportion. Ratio and proportion have already been defined (§ 202). The numerator of the first ratio and the denominator of the second ratio (the first and last terms) are called extremes, the other two terms are called means. The numerators of the two ratios are called antecedents, the denominators are called consequents. The last term of a proportion is called the fourth proportional.

259. Mean Proportion. If the means of a proportion are equal, the proportion is called a mean proportion; the mean is called the mean proportional, or simply the mean, and the last term is called the third proportional.

A continued proportion is a series of equal ratios in which any two successive ratios form a mean proportion.

260. Proportion Proofs. In proving the following proportion theorems, the letters a, b, c, d, may be used to represent like geometrical magnitudes expressed in terms of a common unit of measure. Then, while the ratio may be of two sects, or of two surfaces, that ratio takes a numerical form of expression; as, the ratio of two sects might be .

These proofs must, of course, depend upon the equality axioms.

261. Composition; Division. Four quantities are said to be in proportion by composition when the antecedents become the sums of the terms of the ratios; as,

Four quantities are said to be in proportion by division when the antecedents become the differences of the terms

If the ratios of the sums to the differences are used, the quantities are said to be in proportion by composition

262. Equimultiples. Equimultiples of two quantities are the results obtained by multiplying those quantities by the same number.

NOTE. In proving the theorems in Ratio and Proportion, use the fractional form for the ratios, and use the proportion as an equation; In this way the equality axioms can be used more easily.

as,

a с

b d

THEOREMS

263. Theorem I. If four quantities are in proportion, the product of their means equals the product of their extremes.

264. COR. 1. In a mean proportion, the square of the mean equals the product of the extremes.

265. COR. 2. The value of any term of a proportion can be expressed in the other terms of the proportion.

266. Theorem II. If the product of two quantities equals the product of two other quantities, either pair can be made the means, and the other pair the extremes, of a proportion.

267. COR. 1. If four quantities are in proportion, they are in proportion in any way in which the means of the given proportion are either both means, or both extremes, in the new proportion.

397. Any two sides of a triangle are inversely proportional to the altitudes drawn to them. (Inversely proportional means that one of the ratios is inverted.) Use area formula.

268. Theorem III.

Four quantities which are in proportion are in proportion by composition.

269. Theorem IV. Four quantities which are in proportion are in proportion by division.

270. Theorem V. Four quantities which are in proportion are in proportion by composition and division.

271. Theorem VI. If four quantities are in proportion, equimultiples of the antecedents are in proportion to equimultiples of the consequents.

272. Theorem VII. If four quantities are in proportion, like powers of those quantities are in proportion.

273. Theorem VIII. In a series of equal ratios, the ratio of the sum of the antecedents to the sum of the consequents equals any of the given ratios.

274.

SUMMARY

a

b

d

If a proportion is written in fractional form, and the four terms are considered as forming the vertices of a rectangle, its terms will be in proportion in any order in which the pairs are taken along opposite sides of the rectangle, in the same direction;

that is, both to the right, both down, etc. For example, starting from

Notice that the proportion never goes along a diagonal, as from a to d; this can be kept in mind because the diagonals for a multiplication sign, and diagonal terms can be multiplied but not divided.

The same method applies to composition and to division, the same operations being applied to opposite sides to form the new antecedents, and to form the new consequents. For example, starting b + a d + c

from band going up,

=

This method can be extended to apply to equimultiples, to powers, and to composition and division forms involving equimultiples and powers, and in this way it serves as a test of the correctness of proportion forms.

This is not a proof, but simply a test for correctness, which also acts as a help to the memory by combining all the most important proportion forms in one rule.

SECTION II.

275. Pencil of Lines.

PROPORTIONAL SECTS

Lines that are concurrent are

In the same way a number

spoken of as a pencil of lines. of lines that are all parallel are spoken of as a pencil of parallels.

276. Theorem I. If a line is cut by a pencil of parallels, its sects are proportional to the sects of any other line cut by the same pencil of parallels, including as a special case,

A line parallel to the base of a triangle cuts the sides, or the sides extended, so that the sects are proportional. (Com. case.)

277. COR. 1. A line parallel to the base of a triangle has the same ratio to the base as the lengths it cuts off on the other sides (from their common vertex) have to the whole sides.

278. COR. 2. If parallel lines are cut by a pencil of lines, the sects cut off on the parallels are proportional.

398. In the quadrilateral ABCD, having angle Band angle D right angles, PE and PF are drawn from P in AC perpendicular to BC and DA, respectively. Prove that BE: EC = AF:FD.

399. If BC, of triangle ABC, is extended to X, and AY is cut off on AB equal to CX, then XY is cut by CA in the ratio AB: BC.

400. The diagonals of a trapezoid cut each other proportionally, and their sects are proportional to the bases.

401. If a line cuts the sides of a triangle, extended if necessary, the product of three non-consecutive sects equals the product of the other three sects.