Ex. 236. Touch a given line and touch a given circle at a point P. Ex. 237. Touch a given line AB at P and also touch a given circle. Ex. 238. To inscribe a circle in a given sector. Ex. 239. To construct within a given circle three equal circles, so that each shall touch the other two and also the given circle. Ex. 240. To describe circles about the vertices of a given triangle as centres, so that each shall touch the two others. Ex. 241. To bisect the angle formed by two lines, without producing the lines to their point of intersection. Draw any line EF to BA. Take EG = EH, and produce GH to meet BA at I. Draw the bisector of GI. B D Fi Ex. 242. To draw through a given point P between the sides of an angle BAC a line terminated by the sides of the angle and bisected at P. Ex. 243. Given two points P, Q, and a line AB; to draw lines from P and Q which shall meet on AB and make equal angles with AB. Make use of the point which forms with P a pair of points symmetrical with respect to AB. Ex. 244. To find the shortest path from P to Q which shall touch a line AB. Ex. 245. To draw a common tangent to two given circles. Let r and denote the radii of the circles, O and O' their centres. With centre O and ra respectively, to OA and OC. Draw AB and CD. To draw the internal tangents use an auxiliary O of radius r + r'. PROPORTION. BOOK III. SIMILAR POLYGONS. THE THEORY OF PROPORTION. 323. A proportion is an expression of equality between two equal ratios; and is written in one of the following forms: This proportion is read, "a is to b as c is to d"; or "the ratio of a to b is equal to the ratio of c to d." 324. The terms of a proportion are the four quantities compared; the first and third terms are called the antecedents, the second and fourth terms, the consequents; the first and fourth terms, the extremes, the second and third terms, the means. Thus, in the proportion a : b : c:d; a and c are the antecedents, b and d the consequents, a and d the extremes, b and c the means. 325. The fourth proportional to three given quantities is the fourth term of the proportion which has for its first three terms the three given quantities taken in order. Thus, d is the fourth proportional to a, b, and c in the proportion abc: d. 326. The quantities a, b, c, d, e, are said to be in continued proportion, if a: b = b:c = c:d d: e. = If three quantities are in continued proportion, the second is called the mean proportional between the other two, and the third is called the third proportional to the other two. Thus, in the proportion a:bb: c; b is the mean proportional between a and c; and c is the third proportional to a and b. PROPOSITION I. THEOREM. 327. In every proportion the product of the extremes is equal to the product of the means. 328. The mean proportional between two quantities is equal to the square root of their product. 329. If the product of two quantities is equal to the product of two others, either two may be made the extremes of the proportion in which the other two are made the means. Divide both members of the given equation by bd. PROPOSITION IV. THEOREM. 330. If four quantities are in proportion, they are in proportion by alternation; that is, the first term is to the third as the second is to the fourth. 331. If four quantities are in proportion, they are in proportion by inversion; that is, the second term is to the first as the fourth is to the third. PROPOSITION VI. THEOREM. 332. If four quantities are in proportion, they are in proportion by composition; that is, the sum of the first two terms is to the second term as the sum of the last two terms is to the fourth term. Let To prove that Now Then b that is, Or b с In like manner PROPOSITION VII. THeorem. 333. If four quantities are in proportion, they are in proportion by division; that is, the difference of the first two terms is to the second term as the difference of the last two terms is to the fourth term. |