59. The straight line joining the middle point of the hypotenuse of a right-angled triangle to the right angle is equal to half the hypotenuse. 60. From the angle A of a triangle ABC a perpendicular is drawn to the opposite side, meeting it, produced if necessary, at D; from the angle B a perpendicular is drawn to the opposite side, meeting it, produced if necessary, at E: shew that the straight lines which join D and E to the middle point of AB are equal. 61. From the angles at the base of a triangle perpendiculars are drawn to the opposite sides, produced if necessary: shew that the straight line joining the points of intersection will be bisected by a perpendicular drawn to it from the middle point of the base. 62. In the figure of I. 1, if C and H be the points of intersection of the circles, and AB be produced to meet one of the circles at K, shew that CHK is an equilateral triangle. 63. The straight lines bisecting the angles at the base of an isosceles triangle meet the sides at D and E: shew that DE is parallel to the base. 64. AB, AC are two given straight lines, and P is a given point in the former: it is required to draw through P a straight line to meet AC at Q, so that the angle APQ may be three times the angle AQP. 65. Construct a right-angled triangle, having given the hypotenuse and the sum of the sides. 66. Construct a right-angled triangle, having given the hypotenuse and the difference of the sides. 67. Construct a right-angled triangle, having given the hypotenuse and the perpendicular from the right angle on it. 68. Construct a right-angled triangle, having given the perimeter and an angle. 69. Trisect a right angle. 70. Trisect a given finite straight line. 71. From a given point it is required to draw to two parallel straight lines, two equal straight lines at right angles to each other. 72. Describe a triangle of given perimeter, having its angles equal to those of a given triangle. I. 33, 34. 73. If a quadrilateral have two of its opposite sides parallel, and the two others equal but not parallel, any two of its opposite angles are together equal to two right angles. 74. If a straight line which joins the extremities of two equal straight lines, not parallel, make the angles on the same side of it equal to each other, the straight line which joins the other extremities will be parallel to the first. 75. No two straight lines drawn from the extremities of the base of a triangle to the opposite sides can possibly bisect each other. 76. If the opposite sides of a quadrilateral are equal it is a parallelogram. 77. If the opposite angles of a quadrilateral are equal it is a parallelogram. 78. The diagonals of a parallelogram bisect each other. 79. If the diagonals of a quadrilateral bisect each other it is a parallelogram. 80. If the straight line joining two opposite angles of a parallelogram bisect the angles the four sides of the parallelogram are equal. 81. Draw a straight line through a given point such that the part of it intercepted between two given parallel straight lines may be of given length. 82. Straight lines bisecting two adjacent angles of a parallelogram intersect at right angles. 83. Straight lines bisecting two opposite angles of a parallelogram are either parallel or coincident. 84. If the diagonals of a parallelogram are equal all its angles are equal. 85. Find a point such that the perpendiculars let fall from it on two given straight lines shall be respectively equal to two given straight lines. How many such points are there? 86. It is required to draw a straight line which shall be equal to one straight line and parallel to another, and be terminated by two given straight lines. 87. On the sides AB, BC, and CD of a parallelogram ABCD three equilateral triangles are described, that on BC towards the same parts as the parallelogram, and those on AB, CD towards the opposite parts: shew that the distances of the vertices of the triangles on AB, CD from that on BC are respectively equal to the two diagonals of the parallelogram. 88. If the angle between two adjacent sides of a parallelogram be increased, while their lengths do not alter, the diagonal through their point of intersection will diminish. 89. A, B, C are three points in a straight line, such that AB is equal to BC: shew that the sum of the perpendiculars from A and C on any straight line which does not pass between A and C is double the perpendicular from B on the same straight line. 90. If straight lines be drawn from the angles of any parallelogram perpendicular to any straight line which is Outside the parallelogram, the sum of those from one pair of opposite angles is equal to the sum of those from the other pair of opposite angles. 91. If a six-sided plane rectilineal figure have its opposite sides equal and parallel, the three straight lines joining the opposite angles will meet at a point. 92. AB, AC are two given straight lines; through a given point E between them it is required to draw a straight line GEH such that the intercepted portion GH shall be bisected at the point E. 93. Inscribe a rhombus within a given rhombus, so that one of the angular points of the inscribed figure may bisect a side of the other. 94. ABCD is a parallelogram, and E, F, the middle points of AD and BC respectively; shew that BE and DF will trisect the diagonal AC. I. 35 to 45. ABCD is a quadrilateral having BC parallel to AD; shew that its area is the same as that of the parallelogram which can be formed by drawing through the middle point of DC a straight line parallel to AB. 96. ABCD is a quadrilateral having BC parallel to AD, E is the middle point of DC; shew that the triangle AEB is half the quadrilateral. 97. Shew that any straight line passing through the middle point of the diameter of a parallelogram and terminated by two opposite sides, bisects the parallelogram. 98. Bisect a parallelogram by a straight line drawn through a given point within it. 99. Construct a rhombus equal to a given parallelogram. 100. If two triangles have two sides of the one equal to two sides of the other, each to each, and the sum of the two angles contained by these sides equal to two right angles, the triangles are equal in area. 101. A straight line is drawn bisecting a parallelogram ABCD and meeting AD at E and BC at F: shew that the triangles EBF and CED are equal. 102. Shew that the four triangles into which a parallelogram is divided by its diagonals are equal in area. 103. Two straight lines AB and CD intersect at E, and the triangle AEC is equal to the triangle BED: shew that BC is parallel to AD. 104. ABCD is a parallelogram; from any point P in the diagonal BD the straight lines PA, PC are drawn. Shew that the triangles PAB and PCB are equal. 105. If a triangle is described having two of its sides equal to the diagonals of any quadrilateral, and the included angle equal to either of the angles between these diagonals, then the area of the triangle is equal to the area of the quadrilateral. 106. The straight le which joins the middle points of two sides of any triangle is parallel to the base. 107. Straight lines joining the middle points of adjacent sides of a quadrilateral form a parallelogram. 108. D, E are the middle points of the sides AB, AC of a triangle, and CD, BE intersect at F: shew that the triangle BFC is equal to the quadrilateral ADFE. 109. The straight line which bisects two sides of any triangle is half the base. 110. In the base AC of a triangle take any point D; bisect AD, DC, AB, BC at the points E, F, G, H respectively: shew that EG is equal and parallel to FH. 111. Given the middle points of the sides of a triangle, construct the triangle. 112. If the middle points of any two sides of a triangle be joined, the triangle so cut off is one quarter of the whole. 113. The sides AB, AC of a given triangle ABC are bisected at the points E, F; a perpendicular is drawn from A to the opposite side, meeting it at D. Shew that the angle FDE is equal to the angle BAC. Shew also that AFDE is half the triangle ABC. 114. Two triangles of equal area stand on the same base and on opposite sides: shew that the straight line joining their vertices is bisected by the base or the base produced. 115. Three parallelograms which are equal in all respects are placed with their equal bases in the same straight line and contiguous; the extremities of the base of the first are joined with the extremities of the side opposite to the base of the third, towards the same parts: shew that the portion of the new parallelogram cut off by the second is one half the area of any one of them. 116. ABCD is a parallelogram; from D draw any straight line DFG meeting BC at Fand AB produced at G; draw AF and CG: shew that the triangles ABF, CFG are equal. 117. ABC is a given triangle: construct a triangle of equal area, having for its base a given straight line AD, coinciding in position with AB. 118. ABC is a given triangle: construct a triangle of equal area, having its vertex at a given point in BC and its base in the same straight line as AB. 119. ABCD is a given quadrilateral: construct another quadrilateral of equal area having AB for one side, and for another a straight line drawn through a given point in CD parallel to AB. 120. ABCD is a quadrilateral: construct a triangle whose base shall be in the same straight line as AB, vertex at a given point P in CD, and area equal to that of the given quadrilateral. 121. ABC is a given triangle: construct a triangle of equal area, having its base in the same straight line as AB, and its vertex in a given straight line parallel to AB. 122. Bisect a given triangle by a straight line drawn through a given point in a side. 123. Bisect a given quadrilateral by a straight line drawn through a given angular point. 124. If through the point O within a parallelogram ABCD two straight lines are drawn parallel to the sides, and the parallelograms OB and OD are equal, the point is in the diagonal AC. |