PROPOSITION XLVI. THEOREM. 209. If a figure is symmetrical with respect to two axes perpendicular to each other, it is symmetrical with respect to their intersection as a centre. Let the figure ABCDEFGH be symmetrical with respect to the two axes XX', YY', which intersect at 0. To prove O the centre of symmetry of the figure. Proof. Let N be any point in the perimeter of the figure. Draw NMI to YY', and IKL 1 to XX'. Join LO, ON, and KM. KI= KL, (the figure being symmetrical with respect to XX). KI= OM, Now § 61 But $ 180 (lls comprehended between Ils are equal). .. KLOM, and KLOM is a, § 182 .. LO is equal and parallel to KM. $ 179 In like manner we may prove ON equal and parallel to KM. Hence the points L, O, and N are in the same straight line drawn through the point Ol to KM; and LOON, since each is equal to KM. .. any straight line LON, drawn through O, is bisected at O. .. O is the centre of symmetry of the figure. $ 64 Q. E. D. EXERCISES. 34. The median from the vertex to the base of an isosceles triangle is 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 triangle, formed by producing one of the legs through the vertex, is parallel to the base. 37. State and prove the converse. 38. The altitudes upon the legs of an isosceles triangle are equal. 39. State and prove the converse. 40. The medians drawn to the legs of an isosceles triangle are equal. 41. State and prove the converse. (See Ex. 33.) 42. The bisectors of the base angles of an isosceles triangle are equal. 43. State the converse and the contrary theorems. 44. The perpendiculars dropped from the middle point of the base 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 through the vertex by its own length, the line joining the end of the leg produced to the nearer end of the base is perpendicular to the base. 47. Show that the sum of the interior angles of a hexagon is equal to eight right angles. 48. Show that each angle of an equiangular pentagon is § of a right angle. 49. How many sides has an equiangular polygon, four of whose angles are together equal to seven right angles? 50. How many sides has a polygon, the sum of whose interior angles is equal to the sum of its exterior angles? 51. How many sides has a polygon, the sum of whose interior angles is double that of its exterior angles? 52. How many sides has a polygon, the sum of whose exterior angles is double that of its interior angles? 53. BAC is a triangle having the angle B double the angle A. If BD 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. A E B 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 ▲ COD and AOB are isosceles, and that ▲ AOC and BOD are equal. 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 the ▲ PBG and PBD equal. F G prove 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 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. B FIG. 1. 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. |