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71. In any quadrilateral the squares on the sides are together greater than the squares on its diagonals by four times the square on the line joining the middle points of the diagonals.
72. Divide a line into two parts such the squares on the two parts may be together equal to the square on a given line.
73. If ABC be an isosceles triangle and a perpendicular BD be drawn upon AC : show that the square on the base BC shall be equal to twice the rectangle AC, CD.
74. The square on any straight line drawn from the vertex of an isosceles triangle to the base is less than the square on a side of the triangle by the rectangle contained by the segments of the base. [See Ex. 63, and Prop. 5 Book ii.]
75. If lines be drawn from the angles of a triangle to the middle points of the opposite sides four times the sum of the squares on these lines will be equal to three times the sum of the squares on the sides of the triangle.
76. To draw from two given points two straight lines which shall meet in a given straight line and make equal angles with it.
77. Shew that of any two straight lines that may be drawn from two given points to meet in a given line, the sum is the least when they make equal angles with the line.
78. The three straight lines joining the angular points of a triangle with the middle points of the opposite sides intersect in one point.
79. The exterior angles of a triangle are bisected and the points of intersection of the bisectors is joined with the third angle ; shew that the third angle will be bisected.
80. The quadrilateral figure whose diagonals mutually bisect each other is a parallelogram.
81. In the sides of a square if four points be taken at equal distances from the four angular points taken in order the contained by the straight lines which join them shall also be a square.
82. To construct a triangle having given the middle points of its sides.
83. If the areas of an isosceles triangle and a square be equal the perimeter of the triangle will be the greater.
84. Bisect a triangle by a line drawn parallel to one side.
85. Half the base of a triangle is greater than, equal to, or less than the straight line drawn from the vertex to the middle point of the base, according as the vertical angle is obtuse, right, or acute.
86. If the same straight line bisect the base and the vertical angle the triangle is isosceles.
87. Find a point within a quadrilateral such that the sum of the straight lines drawn from it to the angular points of the figure may be the least possible.
88. Inscribe a square in a right-angled triangle, having one of its angles coinciding with the right angle.
89. Trisect a parallelogram by straight lines drawn from a given point in one of its sides.
90. From a given isosceles triangle to cut off a trapezium which shall have the same base as the triangle, and shall have its three remaining sides equal to each other.
91. To divide a given finite straight line into any given number of equal parts.
92. Two triangles which have the two adjacent sides of a parallelogram for their base, and have their common vertex in the diameter or in the diameter produced shall be equal to one another.
93. The sum of the perpendiculars let fall on the sides of an equilateral triangle from any point within it is constant. 94. Inscribe a square in a given rhombus.
95. Two rectangles have equal areas and equal perimeters : shew that they are equal in all respects.
96. The straight lines AD and BE bisecting the sides BC, AC of a triangle ABC intersect at 0: shew that AO is double of DO.
97. Inscribe an equilateral triangle in a given square.
98. If P is any point in the circumference of a circle ; A and B are points in the diameter equally distant from the centre ; shew that the sum of the squares on AP and BP is constant.
99. Divide a given straight line into two parts, such that the square on one part shall be double the square on the other part.
100. In the figure to Euclid I. 47, shew that the difference of the squares on AD and AE is equal to the difference of the squares on AB and AC.
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