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.

286. Theorem IV.

Two triangles are similar if two

angles of one are equal to two angles of the other.

410. All lines through the point of tangency of two circles are cut proportionally by the circles.

411. If AB is a diameter of a circle, CD a chord perpendicular to AB, then any chord A Y cutting CD at X has the product AX × AY

constant.

412. The product of two sides of a triangle equals the product of the altitude to the third side by the diameter of the circumscribed circle.

413. The product of two sides of a triangle equals the product of the bisector of the included angle by the sect of that bisecting line from the vertex of the angle to the circumscribed circle.

287. Theorem V. Two triangles are similar if two sides of one are proportional to two sides of the other, and the included angles are equal.

414. In any triangle, the orthocenter, the centroid, and the circumcenter, lie in a straight line, and the distance between the first two is double the distance between the second two.

288. Theorem VI. Two triangles are similar if the sides of one are proportional to the sides of the other.

If one had the included angle equal to that of the other, the triangles would be similar; cut it off equal, and show that the triangle obtained is the same triangle as that given.

289. Similar Triangles. Similar triangles are obtained much as congruent triangles were obtained, namely, by three parts. The angles are given equal, but the sides are given proportional instead of equal. Any three parts will do, except two sides proportional and a pair of angles, not included, equal. The following table will serve to show the relation between congruence and similarity.

Similar figures, and especially similar triangles, have many practical applications, such as in finding the height of trees and buildings, and the distances between objects. The principle involved is that if two triangles are similar, any one side, of two pairs of corresponding sides, can be found if the other three are known. Some of the following exercises illustrate this method.

415. How tall is the tree that casts a shadow 50 ft. long at the same time that a pole 6 ft. long casts a shadow 8 ft. long?

416. A building casts a shadow 64 ft. long. A projection on one corner of the building that is found to be 8 ft. from the ground casts a shadow 9 ft. long. How high is the building?

417. It is necessary to find the distance from B to an inaccessible point X. A line BA perpendicular to the sighted line BX is laid off 146 ft. long, and at K, 60 ft. from A on AB, a perpendicular to AB is erected, meeting the sighted line AX at L. If KL is found to be 30 ft. long, how long is BX?

418. Two triangles are similar if the sides of one are parallel to the sides of the other.

419. Two triangles are similar if the sides of one are perpendicular to the sides of the other.

420. If two chords of a circle cut each other, the four sects are proportional.

421. If two secants of a circle intersect, the four sects from the vertex to the circle are proportional.

NOTE. There are now two principal methods by which to find four sects proportional. What are they?

290. Theorem VII. The areas of triangles, or of parallelograms, having an angle of one equal to an angle of the other, have the same ratio as the product of the sides including that angle.

291. COR. 1. The areas of similar triangles have the same ratio as the squares of their corresponding sides.

292. COR. 2. If two triangles that have an angle of one equal to an angle of the other are equivalent, the product of the sides including the angle in one equals the product of the sides including the angle in the other, and conversely.

422. If BC, of triangle ABC, is 8, and CA is 6, how long must CY be, so that a line from Y, on BC, to X, the two thirds point of CA, will cut the triangle into equivalent parts.

423. A line from the midpoint of a side of a triangle must go to what point on a second side to form an equivalent triangle?

424. In a right triangle of legs 3 and 4 ft., the hypotenuse is extended 10 ft. How long must a leg be extended at the same vertex so that the line joining the extremities of the extensions will form a triangle double the given triangle? (Two cases.)

425. If similar triangles are drawn upon the sides of a right triangle as corresponding sides, the triangle on the hypotenuse equals the sum of the other triangles.