Draw A C, CD. The triangles ACB, DCB are similar (II. 23); for they have the angle B common, and by construction AB: ADAD:DB = Therefore, as ACB is isosceles, DCB is also isosceles, and CDCB; therefore also CD DA, and ACD is an isosceles triangle, and the angle AAC D. But the exterior angle BDC=A+ACD=twice the angle A. Therefore, as BBDC, B = twice the angle A. But BACB; therefore the sum of the three angles A, B, and A CB is equal to five times the angle A; or the angle A is one fifth of two right angles, or one tenth of four right angles; therefore the arc BC is one tenth of the circumference, and the chord BC a side of a regular decagon inscribed in the circle. 50. Corollary. By drawing chords joining the alternate angles a regular pentagon will be inscribed. By proceeding as in (46) regular polygons can be inscribed having the number of their sides 20, 40, 80, and so on. PROBLEM XXXVI. 51. To inscribe a regular polygon of fifteen sides in a given circle. Find by (47) the arc AC equal to a sixth of the circumference, and by (49) the arc AB equal to a tenth of the circumference, and the chord BC will be a side of the polygon required. A 52. Corollary. Proceeding as in (46) regular polygons can be inscribed having the number of their sides 30, 60, and so on. PROBLEM XXXVII. 53. To circumscribe about a given circle a polygon similar to a given inscribed regular polygon. Let AD be the given inscribed polygon. Through the points A, B, C, D, E, F draw tangents to the circumference. M These tangents intersecting will form the polygon required. F L G A B H E K Q For the triangles AGB, BHC, &c. are isosceles (19); and as the arcs A B, BC, &c. are equal, the angles GAB, GBA, HBC, HCB, &c. are equal (III. 54); therefore, as the bases A B, BC, &c. are equal, these isosceles triangles are equal. Hence the angles G, H, I, K, L, M are equal, and the polygon MI is equiangular; and as GB=BH HC=CI, &c., GH HI, &c.; therefore the polygon MI is equilateral and regular (II. 32). It is also similar to A D (II. 33); and as its sides are tangents it is circumscribed about the circle. = = 54. Corollary. As (45-52) regular polygons can be inscribed having the number of their sides 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, and so on, regular polygons having the number of their sides represented by these numbers can also be circumscribed about a given circle. EXERCISES. 55. From two given points to draw two equal lines meeting in a given straight line. (I. 53.) 56. Through a given point to draw a line at equal distances from two other given points. 57. From a given point out of a straight line to draw a line making a given angle with that line. (I. 17.) 58. From two given points on the same side of a given line to .draw two lines meeting in the first line and making equal angles with it. 59. From a given point to draw a line making equal angles with the sides of a given angle. 60. Through a given point to draw a line so that the parts of the line intercepted between this point and perpendiculars from two other given points shall be equal. If the three points are in a straight line, the parts equal what? 61. From a point without two given lines to draw a line such that the part between the two lines shall be equal to the part between the given point and the nearer line. When is the Problem impossible? 62. To trisect a right angle. 63. On a given base to construct an isosceles triangle having each of the angles at the base double the third angle. 64. To construct an isosceles triangle when there are given 1st. The base and opposite angle. 2d. The base and an adjacent angle. 3d. A side and an opposite angle. 4th. A side and the angle opposite the base. 65. The base, opposite angle, and the altitude given, to construct the triangle. (III. 22.) (20.) When is the Problem impossible? 66. The base, an angle at the base, and the sum of the sides given, to construct the triangle. When is the Problem impossible? 67. The base, an angle at the base, and the difference of the sides given, to construct the triangle. 1st. When the given angle is adjacent to the shorter side. 2d. When the given angle is adjacent to the longer side. When is the Problem impossible? 68. The base, the difference of the sides, and the difference of the angles at the base given, to construct the triangle. 69. The base, the angle at the vertex, and the sum of the sides given, to construct the triangle. When is the Problem impossible? 70. The base, the angle at the vertex, and the difference of the sides given, to construct the triangle. 71. On a given base to construct a triangle equivalent to a given triangle. 72. With a given altitude to construct a triangle equivalent to a given triangle. 73. Two sides of a triangle and the perpendicular to one of them from the opposite vertex given, to construct the triangle. 74. Two of the perpendiculars from the vertices to the opposite sides and a side given, to construct the triangle. 1st. When one of the perpendiculars falls on the given side. 2d. When neither of the perpendiculars falls on the given side. 75. An angle and two of the perpendiculars from the vertices to the opposite sides given, to construct the triangle. 1st. When one of the perpendiculars falls from the vertex of the given angle. 2d. When neither of the perpendiculars falls from the vertex of the given angle. 76. An angle and the segments of the opposite side made by a perpendicular from the vertex given, to construct the triangle. 77. Given an angle, the opposite side, and the line from the given vertex to the middle of the given side, to construct the triangle. When is the Problem impossible? 78. An angle, a perpendicular from another angle to the opposite side, and the radius of the circumscribed circle given, to construct the triangle. When is the Problem impossible? 79. To divide a triangle into two parts in a given ratio, 1st. By a line drawn from a given point in one of its sides. 80. To trisect a triangle by straight lines drawn from a point within to the vertices. 81. Parallel to the base of a triangle to draw a line equal to the sum of the lower segments of the two sides. 82. Parallel to the base of a triangle to draw a line equal to the difference of the lower segments of the two sides. 83. To inscribe in a given triangle a quadrilateral similar to a given quadrilateral. 84. To divide a given line so that the sum of the squares of the parts shall be equivalent to a given square. 85. To construct a parallelogram when there are given, 1st. Two adjacent sides and a diagonal. 2d. A side and two diagonals. 3d. The two diagonals and the angle between them. 86. To construct a square when the diagonal is given. 87. To construct a parallelogram equivalent to a given triangle and having a given angle. 88. To draw a quadrilateral, the order and magnitude of all the sides and one angle given. Show that sometimes there may be two different polygons satisfying the conditions. 89. To draw a quadrilateral, the order and magnitude of three sides and two angles given. 1st. The given angles included by the given sides. 2d. The two angles adjacent, and one adjacent to the unknown side. 3d. The two angles being opposite each other. 4th. The two angles being both adjacent to the unknown side. In any of these cases can more than one quadrilateral be drawn? 90. To draw a quadrilateral, the order and magnitude of two sides and three angles given. 1st. The given sides being adjacent. 2d. The given sides not being adjacent. |