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39. If equal triangles are described on the same base on opposite sides of it, the line joining their vertices is bisected by the base or base produced.

40. AC, BD are equal straight lines drawn from the extremities of the straight line AB, on opposite sides of it, such that the 4s BAC, ABD are together equal to two right angles, prove that AB bisects the straight line joining CD.

4I. If any point P be taken within a parallelogram ABCD, then ▲ ABP + ▲ CDP = ▲ BCP + ^ DAP.

42. The perimeter of an isosceles triangle is less than that of any other equal triangle on the same base.

43. If an angle of a parallelogram is bisected by a diagonal, then the opposite angle is also bisected by it.

44. If one angle of a parallelogram is bisected by a diagonal, then all the angles are bisected by one or other of the diagonals.

45. If an angle of a parallelogram is bisected by a diagonal, the figure is a rhombus.

46. Any three angles of a convex pentagon with no reentrants are together greater than the angles of a triangle.

47. Any four angles of a convex hexagon with no reentrant angles are together greater than the angles of a quadrilateral.

48. Any angles of a convex rectilineal figure are together greater than all the angles of a figure of which the number of sides is the same as the number of angles of the first figure taken (there being no re-entrants).

49. A quadrilateral figure is divided into four equal triangles by straight lines drawn from a point within the

figure to the angular points; prove that the figure is a parallelogram.

50. Divide a parallelogram into four equal parts by straight lines drawn from a given point in one of its sides.

51. ABDC is a parallelogram; P is a point without the angle BAC; prove that ▲ PAD = ▲ PAB + ▲ PAC.

52. ABDC is a parallelogram; P is a point within the angle BAC; prove that ▲ PAD = the difference between the As PAB, PAC.

53. Two triangles having equal bases are together equal to a triangle having a base equal to that of either of the triangles, and altitude equal to the sum of their altitudes.

54. If one diagonal of a quadrilateral figure bisect the other, it bisects the figure also.

If a quadrilateral, two of whose sides are parallel, be bisected by each of two straight lines drawn from the extremities of the shortest of those parallel sides, then these bisectors bisect each other.

56. If a quadrilateral figure be bisected by a straight line which also bisects its opposite sides, then those opposite sides shall be parallel.

57. The area of a rhombus is equal to half the rectangle contained by the diagonals.

58. Draw an isosceles triangle having each of the angles at the base half as large again as the vertical angle.

59. Bisect a parallelogram by a straight line passing through a given point.

60. Prove that the diagonals of a rhombus bisect one another at right angles.

61. If the opposite angles of a rhombus are together equal to two right angles, the figure is a square.

62. Bisect a triangle by a straight line drawn from a given point in one of its sides.

63. Bisect a quadrilateral by a straight line through one of its angular points.

64. Find a point in a straight line whose distance from another given point in that line shall be equal to its distance from another given straight line.

65. Draw a straight line through a given point which shall make a given angle with a given straight line. Shew that there are two solutions if the given angle is either obtuse or acute.

66. Describe a parallelogram equal in area and perimeter to a given triangle.

67. Describe a triangle equal to a given triangle and having two of its sides respectively equal to two given straight lines. When is the problem impossible?

68. In the base of a triangle find a point from which lines drawn to the sides are equal.

69. Given two sides of a triangle, find the third side so that the area of the triangle may be the greatest possible.

70. Find a point in the side or side produced of a parallelogram, such that the angle which that side makes. with the line joining the point with one extremity of the

opposite side may be bisected by the line joining the point and the other extremity.

71. Inscribe an equilateral triangle in a square, having one of its angular points at one of the corners of the square.

72. Inscribe an equilateral triangle in a square such that one of its angular points shall be at the middle point of one of the sides of the square.

73. Through a given point within a given angle draw a straight line which shall be bisected at that point.

74. The area of a quadrilateral figure whose base is parallel to the opposite side is half that of a parallelogram having the same altitude and base equal to the sum of the parallel sides of the given figure.

75. If one angle of a triangle be equal to the other two together, the greatest side is double of the distance of its middle point from the opposite angle.

76. If each of the equal angles of an isosceles triangle be one-fourth of the third angle, and from one of them a perpendicular be drawn to the base meeting the opposite side produced; then will the part produced, the perpendicular and the remaining side, form an equilateral triangle.

77. Describe a rectangle equal to a given rectangle and having one of its sides equal to a given straight line.

78. Prove that all the interior angles of an octagon are together double all the interior angles of a pentagon.

79. If one quadrilateral figure be equiangular to another and have two adjacent sides equal to the corre

sponding sides of the other, the figure shall be equal in all respects.

80. Two broken lines are composed of straight lines: also the lines and angles of one are respectively equal to the lines and angles of the other; the straight line joining the extremities of the one shall be equal to that joining the extremities of the other.

81. If all the sides and angles of one polygon be given equal to the corresponding sides and angles of another polygon with the exception of one angle and the sides containing it in each respectively, then shall these also be equal.

82. If all the sides and angles of one polygon be given equal to the corresponding sides and angles of another polygon, with the exception of one side in each and the angles at the extremities of those sides, then shall the polygons be equal in all respects.

83. Produce the sides of a given heptagon both ways till they meet forming seven triangles: required the sum of their vertical angles.

84. The sides of a polygon are produced both ways till they meet; find the sum of the vertical angles formed.

85. Describe a square which shall be double of a given square.

86. Describe a square whose area shall be three times that of a given square.

87. Describe two squares equal to a given square: one of them being three times that of the other.

C. G.

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