Then either AB or AC is tangent to BC. For draw OB. ABO, being inscribed in a semicircle, is a right Then angle. Therefore, AB is tangent to BC. (§ 196.) (§ 169.) PROPOSITION XLIV. PROBLEM. 226. Upon a given straight line, to describe a segment which shall contain a given angle. A M E F B Let AB be the given straight line, and A' the given angle. To describe a circumference AMBN, such that every angle inscribed in the segment AMB shall be equal to A'. Construct BAC = ▲ A'. Draw DE perpendicular to AB at its middle point (§ 205). Draw AF perpendicular to AC, meeting DE at 0. With O as a centre, and OA as a radius, describe the circumference AMBN. Then AMB will be the required segment. For, let AGB be any angle inscribed in AMB. Then, AGB is measured by arc ANB. (§ 193.) But AC is tangent to the circle AMB. (§ 169.) Whence, BAC is measured by arc ANB. (§ 197.) Hence, every angle inscribed in the segment AMB is equal to A'. (§ 195.) EXERCISES. 86. Given the middle point of a chord of a circle, to construct the chord. 87. To circumscribe a circle about a given rectangle. 88. To draw a line tangent to a given circle and parallel to a given straight line. 89. To draw a line tangent to a given circle and perpendicular to a given straight line. 90. Through a given point to draw a straight line forming an isosceles triangle with two given intersecting lines. 91. Given the base, an adjacent angle, and the altitude of a triangle, to construct the triangle. 92. Given the base, an adjacent side, and the altitude of a triangle, to construct the triangle. 93. To construct a rhombus, having given its base and altitude. 94. Given the altitude and the sides including the vertical angle of a triangle, to construct the triangle. 95. Given the altitude of a triangle, and the angles at the extremities of the base, to construct the triangle. 96. To construct an isosceles triangle, having given the base and the radius of the circumscribed circle. 97. To construct a square, having given its diagonal. (§ 196.) 98. To construct a right triangle, having given the hypotenuse and the length of the perpendicular drawn to it from the vertex of the right angle. 99. To construct a right triangle, having given the hypotenuse and a leg. 100. Given the base of a triangle and the perpendiculars from its extremities to the other sides, to construct the triangle. 101. To describe a circle of given radius tangent to two given intersecting lines. 102. To describe a circle tangent to a given straight line, having its centre at a given point. 103. To construct a circle having its centre in a given line, and passing through two given points without the line. 104. In a given straight line to find a point equally distant from two given straight lines. 105. Given a side and the diagonals of a parallelogram, to construct the parallelogram. 106. Through a given point within a circle to draw a chord equal to a given chord. (§ 165.) 107. Through a given point to describe a circle tangent to a given straight line at a given point. 108. Through a given point to describe a circle of given radius, tangent to a given straight line. 109. To describe a circle of given radius tangent to two given circles. 110. To describe a circle tangent to two given parallels, and passing through a given point. 111. To describe a circle of given radius, tangent to a given line and a given circle. 112. To construct a parallelogram, having given a side, an angle, and the diagonal drawn from the vertex of the angle. 113. In a given triangle to inscribe a rhombus, having one of its angles coincident with an angle of the triangle. 114. To describe a circle touching two given straight lines, one of them at a given point. 115. In a given sector to inscribe a circle. 116. In a given right triangle to inscribe a square, having one of its angles coincident with the right angle of the triangle. 117. To inscribe a square in a given rhombus. 118. To draw a common tangent to two given circles. 119. Given the base, the altitude, and the vertical angle of a triangle, to construct the triangle. (§ 226.) 120. Given the base of a triangle, its vertical angle, and the median drawn to the base, to construct the triangle. 121. To construct a triangle, having given the middle points of its sides. (§ 130.) 122. Through a vertex of a triangle to draw a straight line equally distant from the other vertices. 123. Given two sides of a triangle, and the median drawn to the third side, to construct the triangle. 124. Given the base, the altitude, and the diameter of the circumscribed circle of a triangle, to construct the triangle. NOTE. For additional exercises on Book II., see p. 224. BOOK III. THEORY OF PROPORTION. - SIMILAR POLYGONS. DEFINITIONS. 227. A Proportion is a statement that two ratios are equal. The statement that the ratio of a to b is equal to the ratio of c to d, may be written in either of the forms 228. The first and fourth terms of a proportion are called the extremes, and the second and third the means. The first and third terms are called the antecedents, and the second and fourth the consequents. Thus, in the proportion a: b c: d, a and d are the extremes, b and c the means, a and c the antecedents, and b and d the consequents. 229. If the means of a proportion are equal, either mean is called a mean proportional between the first and last terms, and the last term is called a third proportional to the first and second terms. Thus, in the proportion a:b = b : C, b is a mean proportional between a and c, and c a third proportional to a and b. 230. A fourth proportional to three quantities is the fourth term of a proportion, whose first three terms are the three quantities taken in their order. Thus, in the proportion a:bc: d, d is a fourth proportional to a, b, and c. PROPOSITION I. THEOREM. 231. In any proportion, the product of the extremes is equal to the product of the means. Let the proportion be a : b = c : d. To prove By § 227, ad= bc. Multiplying both members of the equation by bd, we have 232. COR. The mean proportional between two quantities is equal to the square root of their product. 233. (Converse of Prop. I.) If the product of two quantities is equal to the product of two others, one pair may be made the extremes, and the other pair the means, of a proportion. Dividing both members of the given equation by bd, |