81. To divide a straight line AB into a given number, for example, 5, equal parts. From A draw any line, and along it mark off 5 equal parts. Join the extremity C of the last part with B, and through the other points draw lines parallel to CB. The parts of AB will then be equal (80). A B 82. Two triangles containing the same angles are said to be similar. Similar to" is denoted thus ~. The letters at the vertices of equal angles are put in the same place. Corresponding sides are such which subtend the equal angles; the letters by which they are denoted are placed in the same order, for example, when then A = a; B = b; Cc, and AB, BC, and AC correspond respectively to ab, be and ac. 83. In similar triangles the ratios between two pair of corresponding sides are equal. (The sides are proportional). Place the triangle abc on a small line which is contained in each of them a certain number of times exactly, let this be marked off along them, and be contained, for example, p times in BA and q times in BM; we then have Now if lines be drawn through the points of division parallel to AC, these will divide BC and BN respectively into p and q equal parts (80); therefore we have B) If the lines have no common measure (are incommensurable), divide BM into an indefinite number for example 9 equal parts, and mark off these further; the point A will then fall between two points of division for example the pth and (p+1)th; we then have If we draw parallels as before, we also get be less than I but this fraction can be made as small as we please, for q can be taken as great as we please; but when the difference between the ratios is less than any ever so small quantity, it must be zero, and the ratios therefore be equal*). *) This proof does not hold good here alone, but shews in general, that when two kinds of quantities are proportional when the ratios are commensurable, they will also be proportional when the ratios are incommensurable. In the same way we get, by placing c in C, 84. We have also shewn by this, that a line parallel to one side of a triangle cuts off proportional parts of the two other sides. Conversely: When a line cuts off parts of two sides of a triangle, which are proportional to the sides, then the line will be parallel to the third side. AD AE Hyp. A Th. DE BC. For if DE were not parallel to BC, we would be able to draw another line for example, DFBC; but from that would follow AD AF which is in opposition to what was given. so that this proportion holds, when DE BC, and conversely. 85. To construct a fourth proportional to three given lines, that is, a line which is the fourth term in a proportion, in which the three other terms are the given lines. On the legs of any angle B mark off BM, BA, and BN equal to the given lines; if thereupon MN be joined, and AC drawn parallel to MN, then BC will be the re Α' M B N quired line. If both the mean terms are equal, the required line is called the third proportional to the two given lines and is constructed in the same way. If the given lines are a, b, and c, the required line x, we must have α b с х Here it is of no consequence, whether the letters represent concrete or abstract numbers, expressing the lines measured by the same unit x = bc α it can only be understood in the latter way, as there would be no meaning in multiplying two concrete numbers. Hereby we can again construct x = abc de' by first con this may be expanded to x abcd. efg. line more in the numerator than in the denominator. 86. To divide a given line into parts, which are to one another as given lines or numbers. AB is to be divided into parts, which are to each other as the given lines AD, DE and EC, which are marked off on a line from A; join BC, and draw EG and DF parallel to BC; we then have If numbers are given, any line must be chosen as unit, and this must be marked off as many times as the numbers indicate. 87. A line bisecting an angle of a triangle divides the opposite side into two parts, which are to each other as the sides containing the angle. Α' E B E The line, bisecting the angle adjacent to B, cuts 4C (produced) in a point D1. By putting D, instead of D, all over in the demonstration above, we prove that AB: CB AD1: D1C. We say that D1 divides AC (externally) in the same ratio as D divides AC (internally), or that AC is divided harmonically by D and D1 in the ratio AB: BC. = 1 88. Two figures (systems of points) are said to be similar in the ratio m, when there to every point in the one figure is a corresponding point in the other figure, and the distance between two points in the one figure all over is m times the distance between the corresponding points in the other figure. When m=1, the figures are congruent. That there always is a figure abcd .. .. similar to a given 'figure ABCD.... in a given ratio is shewn thus: From any point O draw lines to the angular points of the given figure, and on these mark off = m. OA; Ob m. OB, &c. d E B Oa a, b, c, d will then be the angular points of the required figure. O is called the centre of similitude, the lines through O rays of similitude. The figures are said to be similarly situated. Of such figures it holds that: Corresponding points lie in the same ray by construction. Corresponding lines (lines joining corresponding points) are parallel, for example, ab For Oa Ob OB AB m. (84) Corresponding lines are proportional in the ratio m: 1, for |