68. ABCD is a parallelogram, E and F the middle points of AD and BC respectively; show that BE and DF will trisect the diagonal AC. 69. If from the diagonal BD of a square ABCD, BE is cut off equal to BC, and EF is drawn perpendicular to BD, show that DE is equal to EF, and also to FC. 70. The bisector of the vertical angle A of a triangle ABC, and the bisectors of the exterior angles at the base formed by producing the sides AB and AC, meet in a point which is equidistant from the base and the sides produced. 71. If the two angles at the base of a triangle are bisected, and through the point of meeting of the bisectors a line is drawn parallel to the base, the length of this parallel between the sides is equal to the sum 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 other, the hypotenuse is double the shortest side. 73. The sum of the perpendiculars dropped from any point in the 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 s, BF the altitude upon AC. Draw PG 1 to BF, and prove the PBG and PBD equal. F E A P B 74. The sum of the perpendiculars dropped from any point within an equilateral triangle to the three sides is constant, and equal to the altitude. HINT. Draw through the point a line II to the base, and apply Ex. 73. 75. What is the locus of all points equidistant from a pair of intersecting lines? 76. In the triangle CAB the bisector of the angle C makes with the perpendicular from C to AB an angle equal to half the difference of the angles A and B. 77. If one angle of an isosceles triangle is equal to 60°, the triangle 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. A. 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. 214. Two circumferences are tangent to each other when they are both tan の B FIG. 1. gent 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. 68. ABCD is a parallelogram, E and F the middle points of AD and BC respectively; show that BE and DF will trisect the diagonal AC. 69. If from the diagonal BD of a square ABCD, BE is cut off equal to BC, and EF is drawn perpendicular to BD, show that DE is equal to EF, and also to FC. 70. The bisector of the vertical angle A of a triangle ABC, and the bisectors of the exterior angles at the base formed by producing the sides AB and AC, meet in a point which is equidistant from the base and the sides produced. 71. If the two angles at the base of a triangle are bisected, and through the point of meeting of the bisectors a line is drawn parallel to the base, the length of this parallel between the sides is equal to the sum 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 other, the hypotenuse is double the shortest side. 73. The sum of the perpendiculars dropped from any point in the 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. A F G E B P 74. The sum of the perpendiculars dropped from any point within an equilateral triangle to the three sides is constant, and equal to the altitude. HINT. Draw through the point a line II to the base, and apply Ex. 73. 75. What is the locus of all points equidistant from a pair of intersecting lines? 76. In the triangle CAB the bisector of the angle C makes with the perpendicular from Cto AB an angle equal to half the difference of the angles A and B. 77. If one angle of an isosceles triangle is equal to 60°, the triangle 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. 214. Two circumferences are tangent to each other when they are both tan Ө B FIG. 1. 68. ABCD is a parallelogram, E and F the middle points of AD and BC respectively; show that BE and DF will trisect the diagonal AC. 69. If from the diagonal BD of a square ABCD, BE is cut off equal to BC, and EF is drawn perpendicular to BD, show that DE is equal to EF, and also to FC. 70. The bisector of the vertical angle A of a triangle ABC, and the bisectors of the exterior angles at the base formed by producing the sides AB and AC, meet in a point which is equidistant from the base and the sides produced. 71. If the two angles at the base of a triangle are bisected, and through the point of meeting of the bisectors a line is drawn parallel to the base, the length of this parallel between the sides is equal to the sum 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 other, the hypotenuse is double the shortest side. 73. The sum of the perpendiculars dropped from any point in the 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. A E F G D P 74. The sum of the perpendiculars dropped from any point within an equilateral triangle to the three sides is constant, and equal to the altitude. HINT. Draw through the point a line II to the base, and apply Ex. 73. 75. What is the locus of all points equidistant from a pair of intersecting lines? 76. In the triangle CAB the bisector of the angle C makes with the perpendicular from C to AB an angle equal to half the difference of the angles A and B. 77. If one angle of an isosceles triangle is equal to 60°, the triangle is equilateral. |