158. AB and AC are tangents to a circle. Through D, any point in the lesser arc BC, is drawn a tangent meeting the first two tangents at E and F. Prove that the perimeter of the triangle AEF is constant, because equal to the sum of AB and AC, and that the angle BOC is twice the angle EOF, O being the centre of the circle. 159. If two opposite sides an inscribed quadrilateral are produced to meet, and a perpendicular be drawn to the bisector of the angle between them from the point of intersection of the diagonals, prove that the perpendicular bisects the angle between the diagonals. 160. A base angle of an inscribed isosceles triangle equals one-third a right angle. Prove that a leg of the triangle equals the radius of the circle. Loci 161. What is the locus of points at a given distance from a given circumference ? a radius distance from a given circumference ? 162. What is the locus of the middle points of a series of parallel chords in a circle ? 163. What is the locus of the middle points of equal chords in a circle ? 164. What is the locus of the middle points of all chords drawn from a given point on the circumference ? 165. A straight line moves so that its two ends constantly touch two indefinite lines which are perpendicular to each other. What is the locus of the middle point of the moving line ? 166. What is the locus of the vertex of the right angle of a right triangle which has a given hypotenuse ? 167. What is the locus of the centres of circles tangent to two intersecting straight lines ? 168. What is the locus of the centres of circles tangent to a given circle at a given point ? BOOK III PROPORTION — SIMILAR POLYGONS REGULAR POLYGONS DEFINITIONS 183. A Proportion is an equality of ratios. Four quantities are said to be in proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth. A proportion may be written in either of the following forms: a:b=c:d, or a:b::c:d, or b d a с 184. The first and fourth terms of a proportion are called the extremes, and the second and third terms the means. The first and third terms are called the antecedents, and the second and fourth terms the consequents. Either ratio is called a couplet. In a proportion in which the means are equal, either mean is called a mean proportional between the other two, and the last term is called a third proportional to the first and second. If the means are unequal, the last term of a proportion is, naturally, a fourth proportional. 185. A Continued Proportion is a series of equal ratios; as, PROPOSITION 186. Theorem. In a proportion the product of the extremes is equal to the product of the means. a:b=c:d or b d ad bc [clearing of fractions] 187. Sch. A proportion is an equation, and many principles used in the transformation of equations, such as squaring the equation, are applicable to proportions. PROPOSITION II 188. Theorem. If the product of two quantities is equal to the product of two others, the factors of either product may be made the extremes, and the factors of the other product the means of a proportion. ad divide by ba] ?] =bc b a or = с =-, etc. divide by what ? PROPOSITION III 189. Theorem. In a proportion the terms are in proportion by Alternation; i.e. the first term is to the third as the second is to the fourth. a:b=c:d [?] [divi :] a с or PROPOSITION IV 190. Theorem. In a proportion the terms are in proportion by Inversion ; i.e. the second term is to the first as the fourth is to the third. 191. Theorem. In a proportion the terms are in proportion by Composition; i.e. in the two couplets, the sums of antecedent and consequent are compared with either the antecedents or with the consequents. PROPOSITION VI 192. Theorem. In a proportion the terms are in proportion by Division; i.e. in the two couplets the differences of antecedent and consequent are compared with either the antecedents or with the consequents. a:b=c:d 193. Cor. In any proportion the terms are in proportion by composition and division. transform and combine a+b:a-b=c+d:c-d results of last two theorems_ [resure PROPOSITION VII 194. Theorem. Both terms of either couplet, or both antecedents, or both consequents, of a proportion may be multiplied or divided by any quantity and the resulting quantities will be in proportion. |