266. Theorem II. If the product of two quantities equa's 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.

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

spoken of as a pencil of lines. In the same way a number 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 B and 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.

279. Points Cutting a Sect; Harmonic Division. A point on a sect, or on the sect extended, is said to cut the sect in the ratio of its distances from the ends of the sect; as, if P is on AB, or on AB extended, the ratio in which it cuts AB is PA: PB. If the point is three fourths as far from A as from B, it cuts AB in the ratio 3: 4, and this is the same whether P is in the sect itself or not.

If two points cut the same sect in the same ratio, one internally, the other externally, they are said to cut the sect harmonically. The equal ratios must, of course, be taken from corresponding ends of the sect; as, if P and Q cut AB harmonically, PA: PB = QA : QB.

Notice that a sect cannot be cut externally in the ratio 1, for if P is in the extension of AB, PA cannot equal PB.

280. Theorem II. A sect can be cut in the same ratio internally by but one point, and externally by but one point.

402. The interior common tangents of two circles (those between the circles) meet the center line at the same point.

403. The exterior common tangents of two circles meet the center line at the same point.

404. The interior and exterior common tangents to two circles cut the center sect harmonically.

405. A line through the ends of two parallel radii of two circles meets the center line at the same point as the common tangents.

NOTE. The points where the common tangents meet the center line are called the inverse, and direct centers of similitude.

406. If two points cut a sect harmonically, they include a second sect, which is cut harmonically by the ends of the first sect.

281. Theorem III. A line that cuts two sides of a triangle proportionally is parallel to the third side.

407. If a sect joins the one third points of two sides of a triangle (taken from their common vertex), what part of the third side is it?

SECTION III. SIMILAR FIGURES

282. Similar Figures. Polygons are said to be similar if their corresponding angles are equal and their corresponding sides are proportional. See also Appendix, § 349.

There are now three things which can be proved about polygons: that they are congruent, equivalent, or similar. Equivalent means of the same size (as regards surface), similar means of the same shape, while congruent includes both size and shape. Notice that the sign for congruent is composed of the equivalent sign and the similar sign. These facts are not definitions of the words, but serve to show the distinction in meaning in a somewhat different light.

*283. Polygons similar to the same polygon are similar to each other.

*284. Perimeters of similar polygons are proportional to any pair of corresponding sides.

*285. Regular polygons of the same number of sides are similar.

408. If two similar polygons are placed with a pair of corresponding sides parallel (the polygons lying on the same sides of those lines), the lines through the corresponding pairs of vertices will form a pencil, which is cut proportionally by the vertices of the polygon.

409. If two polygons lie in a pencil of lines, and their vertices cut the lines proportionally, the polygons are similar.

SMITH'S SYL. PL. GEOM.—10 145