EXAMPLES. 139. In a right-angled triangle the sides containing the right angle are a and b, their projections on the hypothenuse a and B, the altitude p, and the hypothenuse h; how great are the other parts, when as +63 Va 3a-2b' when a, b, and c are given lines. 141. In a triangle, the sides of which are a transversal, d= 1.68, is drawn parallel to the side c; how great are the parts in which it divides the sides a and b? 142. In a rectangle the one side is 6 feet, and the diagonal is 2 feet longer than the other side. How great is this? 143. In a triangle the one altitude is 15, and the parts of the base 6 and 10. At what distance over the base is the point of intersection of the altitudes? 144. How great is the diagonal of a square with the side a? 145. In a right-angled triangle the hypothenuse is m2+n2 and the one side 2mn. Find the other side. 146. How great is the altitude of an equilateral triangle with the side a? 147. In a quadrilateral the diagonals are at right angles to each other. Prove that the sum of the squares on the one pair of opposite sides is equal to the sum of the squares on the other pair. 148. A circle passes through the centre C of another circle and touches it at A. A line perpendicular to AC cuts the first circle in D, the second in E. 149. From a point E in the circumference of a circle a perpendicular EP is dropped on a diameter. D is the middle point of the semicircle with radius r. Shew that 150. In a circle with radius r an angle of 45° is drawn at the centre. On the one leg of the angle a perpendicular CE is raised, cutting the other leg in D and the circle in E. 151. From a point to a circle a secant is drawn, both parts of which equal a. How great is the tangent from the same point? 152. In a right-angled triangle with the sides containing the right angle equal to 1.41 and 1.88, a perpendicular is raised on the middle of the hypothenuse. How great are the parts in which it divides the one side? 153. In a semicircle there is over each half r of the diameter described other semicircles. How great is the radius of the circle touching the three semicircles? 154. In a triangle with sides a, b, and c, a perpendicular is dropped on c from the opposite vertex. Prove that a2b22cl, when I is the distance from the foot of the perpendicular to the middle point of c, and when a> b. Prove that a2 + b2 {{ c2 + 2m2. (m, the median line). = 155. A and B are two given points in a diameter, equidistant from the centre, P is any point in the circumference. AP+BP Shew that AP + BP is constant. 156. Through A (Ex. 155) any chord is drawn, the extremities of which are joined to B. Shew that the sum of the squares on the sides of the triangle thus formed is constant. 157. In a triangle with sides a, b, and c, the angle between the two first is bisected. How great are the parts in which the bisecting line divides the third side? B) If the lines have no common measure (are incommensurable), divide BM into an indefinite number for example q 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 I be less than ; 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. AE AC A AB 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. According to a proposition in proportion, the proportion 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 M B N Α' ୯ 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 Here α b C х it is of no consequence, whether the letters represent concrete or abstract numbers, expressing the lines measured by the same unit X bc a 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 a abc - by first con de 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. |