10 34. The median from the vertex to the base of an isoscele perpendicular to the base, and bisects the vertical angle. 35. State and prove the converse. 36. The bisector of an exterior angle of an isosceles trian 37. State and prove the converse. 38. The altitudes upon the legs of an isosceles triangle ar 39. State and prove the converse. 40. The medians drawn to the legs of an isosceles triangl 41. State and prove the converse. (See Ex. 33.) 42. The bisectors of the base angles of an isosceles triangl 43. State the converse and the contrary theorems. 44. The perpendiculars dropped from the middle point of an isosceles triangle upon the legs are equal. 45. State and prove the converse. 46. If one of the legs of an isosceles triangle is produced vertex by its own length, the line joining the end of the leg the nearer end of the base is perpendicular to the base. 47. Show that the sum of the interior angles of a hexagon eight right angles. 48. Show that each angle of an equiangular pentagon is angle. 49. How many sides has an equiangular polygon, four of w are together equal to seven right angles? 50. How many sides has a polygon, the sum of whose int is equal to the sum of its exterior angles ? 51. How many sides has a polygon, the sum of whose int is double that of its exterior angles? 52. How many sides has a polygon, the sum of whose ext is double that of its interior angles? bisect the angle B, and meet AC in D, show that BD is equal to AD. 54. If from any point in the base of an isosceles triangle parallels to the legs are drawn, show that a parallelogram is formed whose perimeter is constant, and equal to the sum of the legs of the triangle. 55. The lines joining the middle points of the sides of a triangle divide the triangle into four equal triangles. 56. The lines joining the middle points of the side of a square, taken in order, enclose a square. 57. The lines joining the middle points of the sides of a rectangle, taken in order, enclose a rhombus. 58. The lines joining the middle points of the sides of a rhombus, taken in order, enclose a rectangle. 59. The lines joining the middle points of the sides of an isosceles trapezoid, taken in order, enclose a rhombus or a square. 60. The lines joining the middle points of the sides of any quadrilateral, taken in order, enclose a parallelogram. 61. The median of a trapezoid passes through the middle points of the two diagonals. 62. The line joining the middle points of the diagonals of a trapezoid is equal to half the difference of the bases. 63. In an isosceles trapezoid each base makes equal angles with the legs. HINT. Draw CE | DB. 64. In an isosceles trapezoid the opposite angles are supplementary. 65. If the angles at the base of a trapezoid are equal, the other angles are equal, and the trapezoid is isosceles. 66. The diagonals of an isosceles trapezoid are equal. 67. If the diagonals of a trapezoid are equal, the trapezoid is isosceles. HINT. Draw CE and DF 1 to CD. Show that A ADF and BCE are equal, that A COD and AOB are 1 DOD 68. ABCD is a parallelogram, E and F the middle points BC respectively; show that BE and DF will trisect the diag 69. If from the diagonal BD of a square ABCD, BE is cu to BC, and EF is drawn perpendicular to BD, show that D to EF, and also to FC. 70. The bisector of the vertical angle A of a triangle AB bisectors of the exterior angles at the base formed by producin AB and AC, meet in a point which is equidistant from the ba sides produced. 71. If the two angles at the base of a triangle are bis through the point of meeting of the bisectors a line is drawn the base, the length of this parallel between the sides is equal of the segments of the sides between the parallel and the base 72. If one of the acute angles of a right triangle is double the hypotenuse is double the shortest side. 73. The sum of the perpendiculars dropped from any po base of an isosceles triangle to the legs is constant, and equal to the altitude upon one of the legs. HINT. Let PD and PE be the two Is, BF the altitude upon AC. Draw PG 1 to BF, and prove the PBG and PBD equal. Α E F 74. The sum of the perpendiculars dropped from any point equilateral triangle to the three sides is constant, and equ altitude. HINT. Draw through the point a line II to the base, and ap 75. What is the locus of all points equidistant from a pai secting lines? 76. In the triangle CAB the bisector of the angle C make perpendicular from C to AB an angle equal to half the differ angles A and B. 77. If one angle of an isosceles triangle is equal to 60°, tl is equilateral. BOOK II. THE CIRCLE. DEFINITIONS. 210. A circle is a portion of a plane bounded by a curved line called a circumference, all points of which are equally distant from a point within called the centre. 211. A radius is a straight line drawn from the centre to the circumference; and a diameter is a straight line drawn through the centre, having its extremities in the circumference. By the definition of a circle, all its radii are equal. All its diameters are equal, since the diameter is equal to two radii. 212. A secant is a straight line which intersects the circumference in two points; as, AD, Fig. 1. 213. A tangent is a straight line which touches the circumference but does not intersect it; as, BC, Fig. 1. The point in which the tangent touches the circumference is called the point of contact, or point of tangency. Ө B FIG. 1. D 214. Two circumferences are tangent to each other when they are both tangent to a straight line at the same point; and are tangent internally or externally, according as one circumference lies wholly within or without the other. An arc equal to one-half the circumference is calle circumference. 216. A chord is a straight line having its extremit circumference. Every chord subtends two arcs whose sum is th ference; thus, the chord AB (Fig. 3) subtends the sn AB and the larger arc BCDEA. If a chord and i spoken of, the less arc is meant unless it is otherwise. 217. A segment of a circle is a portion of a circle by an arc and its chord. A segment equal to one-half the circle is called a se 218. A sector of a circle is a portion of the circle by two radii and the arc which they intercept. A sector equal to one-fourth of the circle is called a 219. A straight line is inscribed in a circle if it is a 220. An angle is inscribed in a circle if its vertex circumference and its sides are chords. 221. An angle is inscribed in a segment if its ve the arc of the segment and its sides pass through th ities of the arc. 222. A polygon is inscribed in a circle if its chords of the circle. 223. A circle is inscribed in a polygon if the circu touches the sides of the polygon but does not interse |