COROLLARIES. 1. If two polygons are similar, they are mutually equiangular and their corresponding sides are proportional. For if placed in perspective as on p. 163, 1. OA1 OA2 OB1: OB2. 2... A1B1 | A2 B2, and so for other sides. = ..... 4. Also A1B1: A2 B2 = B10: B2O = B1C1: B2C2 = Def. sim. figs. Th. 3 I, th. 17, cor. 5 Th. 1, cor. 2 2. Polygons similar to the same polygon are similar to each other. For they have angles equal to those of the third polygon, and the ratios of their sides equal the ratios of the sides of the third polygon. 3. The perimeters of similar polygons have the same ratio as the corresponding sides. For by cor. 1, A1B1: A2B2 = B1C1: B2C2 | .. ZA1B1:ΣA2B2 =r (Why?); that is, the perimeters have the ratio r, which is the ratio of the corresponding sides. 4. Two similar polygons can be divided into the same number of triangles similar each to each, and similarly placed. For O and O' coincide, the figures can be placed having O within each, and the triangles A1OB1, A2OB2 are similar, by th. 8. Theorem 13. In a right-angled triangle the perpendicular from the vertex of the right angle to the hypotenuse divides the triangle into two triangles which are similar to the whole and to each other. 3. ...A ACDA ABC, which proves (1). Th. 8 4. Similarly, A CBD ▲ ABC, which proves (2). Th. 8 5. COROLLARIES. 1. Either side of a right-angled triangle is the mean proportional between the hypotenuse and its segment adjacent to that side. For from step 3, AB AC AC AD; and from 4, AB : BC = = BC: DB. 2. The perpendicular from the vertex of the right angle to the hypotenuse is the mean proportional between the segments of the hypotenuse. For from step 5, AD: CD CD: DB. EXERCISES. 430. Converse of th. 13: If the perpendicular drawn from the vertex of a triangle to the base is the mean proportional between the segments of the base, the triangle is right-angled. 431. Any chord of a circle is the mean proportional between its projection on the diameter from one of its extremities, and the diameter itself. 432. In the figure on p. 164, if AD represents three units, and DB represents one unit, what number is represented by CD? Draw the figure accurately with rule and compasses, and measure CD as closely as possible. Show that, in similar manner, √√5, √7, etc., can be found. 433. If a perpendicular is let fall from any point on a circumference, to any diameter, it is the mean proportional between the segments into which it divides that diameter. 434. If a point P, within a circle of radius r, is joined to the center of that circle by a line a, and if from P a perpendicular h is drawn to a and terminated by the circumference, show that h2 = r2 — a2 = (r + a) (r — a), and that this proves cor. 2 and ex. 433. 435. If two fixed parallel tangents are cut by a variable tangent, the rectangle of the segments of the latter is constant. Prove that 436. Through any point in the common chord of two intersecting circumferences two chords are drawn, one in each circle. the four extremities of these chords are concyclic. 437. If the bisectors of the interior and exterior angles at B, in the figure of th. 13, meet b at F and E, respectively, then BC is the mean proportional between FC and CE. Problem 1. To divide a line-segment into parts proportional to the segments of a given line. Given the line OX', and the line A' B' C' oblique to each other at a common end point O, draw XX'. Post. 2 2. From A, B, draw lines II XX', cutting OX' at A', B',...... ..... I, pr. 6 Proof. ... OX, OX' are two transversals of a pencil of Ils, the corresponding segments are in proportion. 3. Then OX' is divided as required. Th. 1 COROLLARIES. 1. A given line can be divided into parts proportional to any number of given lines. For that number of given lines may be laid off as OA, AB, BC,..... on OX. 2. A line can be divided into any number of equal parts. NOTE. While a straight line can be divided into any number of equal parts, by means of the straight ruler and the compasses, a circumference cannot be divided into 7, 9, 11, 13, and, in general, any prime number of equal parts beyond 5. The exceptions are noted in Book V. Problem 2. To find the fourth proportional to three given lines. Given three lines, a, b, c. Required to find x such that' abc: x. a X Construction. Proof. 1. From the vertex of a pencil of two lines, with the compasses lay off a, b, in order, on one line, and c on the other line. 2. Join the end-points of a, c, remote from the vertex, by I. Post. 2 3. From the end-point of b, remote from a, draw a line parallel to l. 4. This will cut off x, the line required. I, pr. 6 DEFINITION. If a: b = b x, x is called the third proportional to a and b. COROLLARY. be found. The third proportional to two given lines can For to find x such that a: b == EXERCISE. b x, make cb in the above solution. 438. The problem admits of a considerable variation of the figure, as suggested by the figure given in ex. 400. Invent another solution from this suggestion. Problem 3. To find the mean proportional between two given lines. 2. With center O and radius OB, describe a circle. 3. From D draw DC AB, to meet circumference at C. Post. 6 I, pr. 2 Then CD is the mean proportional. Proof. Th. 13, cor. 2 DEFINITION. A line is said to be divided in extreme and mean ratio by a point when one of the segments is the mean proportional between the whole line and the other segment. Thus AB is divided internally in extreme and mean ratio at P, if P AB: AP AP : PB; and externally in such ratio at P', if AB : AP' = AP' : P'B. P. B It will be seen that AP2 AB PB, this is simply the "Golden Section or the "Median Section" of Book II, pr. 6. In fact, this division of a line is known by all three names. A more practical solution of the problem of the Golden Section than that given in Book II can now be given. Describe a O with center C and radius CB. Draw AC cutting the circumference in X and Y. Describe two arcs with center A and radii AX and AY, thus fixing points P, P'. .. AB is divided internally at P and externally at P' in Golden Section. It should be noticed that if the sense of the lines is considered (that is, considering APPA), the above solutions would be identical if X and Y were interchanged, and P' substituted for P. |