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other side of AB: prove the points F, C', G to be in the same straight line; and the figure CFG to be an equilateral triangle.

57. ABC is a triangle and the exterior angles at B and C are bisected by lines BD, CD respectively, meeting in D: shew that the angle BDC and half the angle BAC make up a right angle.

58. If the exterior angle of a triangle be bisected, and the angles of the triangle made by the bisectors be bisected, and so on, the triangles so formed will tend to become eventually equilateral.

59. If in the three sides AB, BC, CA of an equilateral triangle ABC, distances AE, BF, CG be taken, each equal to a third of one of the sides, and the points E, F, G be respectively joined (1) with each other, (2) with the opposite angles: shew that the two triangles so formed, are equilateral triangles.

IV.

60. Describe a right-angled triangle upon a given base, having given also the perpendicular from the right angle upon the hypotenuse.

61. Given one side of a right-angled triangle, and the difference between the hypotenuse and the sum of the other two sides, to construct the triangle.

62. Construct an isosceles right-angled triangle, having given (1) the sum of the hypotenuse and one side; (2) their difference.

63. Describe a right-angled triangle of which the hypotenuse and the difference between the other two sides are given.

64. Given the base of an isosceles triangle, and the sum or difference of a side and the perpendicular from the vertex on the base. Construct the triangle.

65. Make an isosceles triangle of given altitude whose sides shall pass through two given points and have its base on a given straight line.

66. Construct an equilateral triangle, having given the length of the perpendicular drawn from one of the angles on the opposite side. 67. Having given the straight lines which bisect the angles at the base of an equilateral triangle, determine a side of the triangle.

68. Having given two sides and an angle of a triangle, construct the triangle, distinguishing the different cases.

69. Having given the base of a triangle, the difference of the sides, and the difference of the angles at the base; to describe the triangle. 70. Given the perimeter and the angles of a triangle, to construct it.

71. Having given the base of a triangle, and half the sum and half the difference of the angles at the base; to construct the triangle.

72. Having given two lines, which are not parallel, and a point between them; describe a triangle having two of its angles in the respective lines, and the third at the given point; and such that the sides shall be equally inclined to the lines which they meet.

73. Construct a triangle, having given the three lines drawn from the angles to bisect the sides opposite.

74. Given one of the angles at the base of a triangle, the base itself, and the sum of the two remaining sides, to construct the triangle.

75. Given the base, an angle adjacent to the base, and the difference of the sides of a triangle, to construct it.

76. Given one angle, a side opposite to it, and the difference of the other two sides; to construct the triangle.

77. Given the base and the sum of the two other sides of a triangle, construct it so that the line which bisects the vertical angle shall be parallel to a given line.

V.

78. From a given point without a given straight line, to draw a line making an angle with the given line equal to a given rectilineal angle. 79. Through a given point A, draw a straight line ABC meeting two given parallel straight lines in B and C, such that BC may be equal to a given straight line.

80. If the line joining two parallel lines be bisected, all the lines drawn through the point of bisection and terminated by the parallel lines are also bisected in that point.

81. Three given straight lines issue from a point: draw another straight line cutting them so that the two segments of it intercepted between them may be equal to one another.

82. AB, AC are two straight lines, B and C given points in the same; BD is drawn perpendicular to AC, and DE perpendicular to AB; in like manner CF is drawn perpendicular to AB, and FG to AC. Shew that EG is parallel to BC.

83. ABC is a right-angled triangle, and the sides AC, AB are produced to D and F; bisect FBC and BCD by the lines BE, CE, and from E let fall the perpendiculars EF, ED. Prove (without assuming any properties of parallels) that ADEF is a square.

84. Two pairs of equal straight lines being given, shew how to construct with them the greatest parallelogram.

85. With two given lines as diagonals describe a parallelogram which shall have an angle equal to a given angle. Within what limits must the given angle lie?

86. Having given one of the diagonals of a parallelogram, the sum of the two adjacent sides and the angle between them, construct the parallelogram.

87. One of the diagonals of a parallelogram being given, and the angle which it makes with one of the sides, complete the parallelogram, so that the other diagonal may be parallel to a given line.

88. ABCD, ABCD are two parallelograms whose corresponding sides are equal, but the angle A is greater than the angle A', prove that the diameter AC is less than A'C', but BD greater than B'D.

89. If in the diagonal of a parallelogram any two points equidistant from its extremities be joined with the opposite angles, a figure will be formed which is also a parallelogram.

90. From each angle of a parallelogram a line is drawn making

the same angle towards the same parts with an adjacent side, taken always in the same order; shew that these lines form another parallelogram similar to the original one.

91. Along the sides of a parallelogram taken in order, measure AA'= BB' CC DD': the figure A'B'C'D' will be a parallelogram.

92. On the sides AB, BC, CD, DA, of a parallelogram, set off AE, BF, CG, DH, equal to each other, and join AF, BG, CH,DE: these lines form a parallelogram, and the difference of the angles AFB, BGC, equals the difference of any two proximate angles of the two parallelograms.

93. OB, OC are two straight lines at right angles to each other, through any point P any two straight lines are drawn intersecting OB, OC, in B, B, C, C', respectively. If D and D' be the middle points of BB and CC, shew that the angle B'PD' is equal to the angle DOD.

94. ABCD is a parallelogram of which the angle C'is opposite to the angle A. If through A any straight line be drawn, then the distance of C is equal to the sum or difference of the distances of B and of D from that straight line, according as it lies without or within the parallelogram.

95. Upon stretching two chains AC, BD, across a field ABCD, I find that BD and AC make equal angles with DC, and that AC makes the same angle with AD that BD does with BC; hence prove that AB is parallel to CD.

96. To find a point in the side or side produced of any parallelogram, such that the angle it makes with the line joining the point and one extremity of the opposite side, may be bisected by the line joining it with the other extremity.

97. When the corner of the leaf of a book is turned down a second time, so that the lines of folding are parallel and equidistant, the space in the second fold is equal to three times that in the first.

VI.

98. If the points of bisection of the sides of a triangle be joined. the triangle so formed shall be one-fourth of the given triangle.

99. If in the triangle ABC, BC be bisected in D, AD joined and bisected in E, BE joined and bisected in F, and CF joined and bisected in G; then the triangle EFG will be equal to one-eighth of the triangle ABC.

100. Shew that the areas of the two equilateral triangles in Prob. 59, p. 78, are respectively, one-third and one-seventh of the area of the original triangle.

101. To describe a triangle equal to a given triangle, (1) when the base, (2) when the altitude of the required triangle is given.

102. To describe a triangle equal to the sum or difference of two given triangles.

103. Upon a given base describe an isosceles triangle equal to a given triangle.

104. Describe a right-angled triangle equal to a given triangle ABC.

105. To a given straight line apply a triangle which shall be equal

to a given parallelogram and have one of its angles equal to a given rectilineal angle.

106. Transform a given rectilineal figure into a triangle whose vertex shall be in a given angle of the figure, and whose base shall be in one of the sides.

107. Divide a triangle by two straight lines into three parts which when properly arranged shall form a parallelogram whose angles are of a given magnitude.

108. Shew that a scalene triangle cannot be divided into two parts which will coincide.

109. If two sides of a triangle be given, the triangle will be greatest when they contain a right angle.

110. Of all triangles having the same vertical angle, and whose bases pass through a given point, the least is that whose base is bisected in the given point.

111. Of all triangles having the same base and the same perimeter, that is the greatest which has the two undetermined sides equal.

112. Divide a triangle into three equal parts, (1) by lines drawn from a point in one of the sides: (2) by lines drawn from the angles to a point within the triangle: (3) by lines drawn from a given point within the triangle. In how many ways can the third case be done?

113. Divide an equilateral triangle into nine equal parts.

114. Bisect a parallelogram, (1) by a line drawn from a point in one of its sides: (2) by a line drawn from a given point within or without it: (3) by a line perpendicular to one of the sides: (4) by a line drawn parallel to a given line.

115. From a given point in one side produced of a parallelogram, draw a straight line which shall divide the parallelogram into two equal parts.

116. To trisect a parallelogram by lines drawn (1) from a given point in one of its sides, (2) from one of its angular points.

VII.

117. To describe a rhombus which shall be equal to any given quadrilateral figure.

118. Describe a parallelogram which shall be equal in area and perimeter to a given triangle.

119. Find a point in the diagonal of a square produced, from which if a straight line be drawn parallel to any side of the square, and meeting another side produced, it will form together with the produced diagonal and produced side, a triangle equal to the square.

120. If from any point within a parallelogram, straight lines be drawn to the angles, the parallelogram shall be divided into four triangles, of which each two opposite are together equal to one-half of the parallelogram.

121. If ABCD be a parallelogram, and E any point in the diagonal AC, or AC produced; shew that the triangles EBC, EDC, are equal, as also the triangles EBA and EBD.

122. ABCD is a parallelogram, draw DFG meeting BC in F,

and AB produced in G; join AF, CG; then will the triangles A.B.F, CFG be equal to one another.

123. ABCD is a parallelogram, E the point of intersection of its diagonals, and K any point in AD. If KB, KC be joined, shew that the figure BKEC is one-fourth of the parallelogram.

124. Let ABCD be a parallelogram, and O any point within it, through O draw lines parallel to the sides of ABCD, and join 04, OC; prove that the difference of the parallelograms DO, BŎ is twice the triangle (AC.

125. The diagonals AC, BD of a parallelogram intersect in O, and P is a point within the triangle AOB; prove that the difference of the triangles APB, CPD is equal to the sum of the triangles APC, BPD.

126. If K be the common angular point of the parallelograms about the diameter AC (fig. Euc. I. 43.) and BD be the other diameter, the difference of these parallelograms is equal to twice the triangle BKD.

127. The perimeter of a square is less than that of any other parallelogram of equal area.

128. Shew that of all equiangular parallelograms of equal perimeters, that which is equilateral is the greatest.

129. Prove that the perimeter of an isosceles triangle is greater than that of an equal right-angled parallelogram of the same altitude.

VIII.

130. If a quadrilateral figure is bisected by one diagonal, the second diagonal is bisected by the first.

131. If two opposite angles of a quadrilateral figure are equal, shew that the angles between opposite sides produced are equal.

132. Prove that the sides of any four-sided rectilinear figure are together greater than the two diagonals.

133. The sum of the diagonals of a trapezium is less than the sum of any four lines which can be drawn to the four angles, from any point within the figure, except their intersection.

134. The longest side of a given quadrilateral is opposite to the shortest; shew that the angles adjacent to the shortest side are together greater than the sum of the angles adjacent to the longest side.

135. Give any two points in the opposite sides of a trapezium, inscribe in it a parallelogram having two of its angles at these points.

136. Shew that in every quadrilateral plane figure, two parallelograms can be described upon two opposite sides as diagonals, such that the other two diagonals shall be in the same straight line and equal. 137. Describe a quadrilateral figure whose sides shall be equal to four given straight lines. What limitation is necessary ?

138. If the sides of a quadrilateral figure be bisected and the points of bisection joined, the included figure is a parallelogram, and equal in area to half the original figure.

139. A trapezium is such, that the perpendiculars let fall on a diagonal from the opposite angles are equal. Divide the trapezium into four equal triangles, by straight lines drawn to the angles from a point within it.

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