BOOK V.

96. Between two given straight lines to draw a straight line equal to one given straight line and parallel to another.

97. To find a point at given distances from two given lines.

98. Through a given point between two straight lines to draw a line such that the part between the given lines shall be bisected at the given point.

99. To a straight line from two given points on opposite sides of it to draw lines forming an angle that is bisected by the given line.

100. From a given point to draw two straight lines, making respectively equal angles with two given intersecting straight lines.

101. Between two intersecting lines to place a given straight line so that it shall make equal angles with each.

102. In a given straight line to find a point equally distant from two given lines.

When is the problem impossible?

103. In a triangle A B C to draw from a given point D, in the side A B, or the side produced, a straight line to A C so that it shall be bisected by BC.

104. In the sides of a triangle to find a point from which lines drawn parallel to the other sides and limited by them are equal.

105. To draw a line parallel to one of the sides of a triangle so that the part intercepted between the other two sides shall be equal to the difference between one of these sides and the side to which the required line is parallel.

106. Draw two unequal triangles that have a side and two angles of one equal to a side and two angles of the other.

107. From a given isosceles triangle to cut off a trapezoid having for its base the base of the triangle and the other three sides equal to each other.

108. A side and two medial lines of a triangle given, to construct the triangle.

1st. When the given medial lines are from the extremities of the given side.

2d. When only one of the medial lines is from the extremity of the given side.

109. Two sides and one medial line given, to construct the triangle.

1st. When the medial line is from the vertex of the two given

sides.

2d. When the medial line is not from the vertex of the two given sides.

110. The three medial lines given, to construct the triangle.

111. A side of an isosceles triangle and the sum of the perpendiculars from any point of the base to the opposite sides given, to construct the triangle.

112. The three perpendiculars from any point within to the sides of an equilateral triangle given, to construct the triangle.

113. The three lines from any point within an equilateral triangle to the vertices given, to construct the triangle.

114. The three altitudes of a triangle given, to construct the triangle.

115. The base, the sum of the sides, and the difference of the angles at the base given, to construct the triangle. (II. 68.)

116. Two angles and the sum of two sides given, to construct the triangle.

117. Two angles and the perimeter given, to construct the triangle.

118. An angle, its bisector, and the perpendicular from its vertex to the opposite side given, to construct the triangle.

119. An angle, the medial line and the perpendicular from the vertex of the given angle to the opposite side given, to construct the triangle.

120. The perpendicular, the bisector, and the medial line, from the same vertex given, to construct the triangle.

121. The feet of the perpendiculars from the vertices to the opposite sides given, to construct the triangle.

122. The middle points of the sides of a triangle given, to construct the triangle.

123. An angle, the angle between the bisector and the perpendicular from the vertex of the given angle to the opposite side, and the perpendicular given, to construct the triangle.

124. Two sides and the difference of the segments of the base made by a perpendicular from the vertical angle to the base given, to construct the triangle.

Is there any ambiguity in this Problem?

125. The base, the foot of the perpendicular from the vertex to the base, and the sum, or the difference, of the other two sides given, to construct the triangle.

126. The base and the sum of the two other sides given, to construct the triangle so that the bisector of the vertical angle shall be parallel to a given line.

127. The sum of the base and perpendicular, of the base and hypothenuse, of the perpendicular and hypothenuse of a right triangle given, to construct the triangle.

BOOK VI.

82. If straight lines are parallel the intersections of any planes passing through these lines are parallel.

83. The projections of parallel lines on any plane are parallel.

84. If parallel planes cut two planes not parallel, the angle of the intersections of one of these parallel planes with the planes not parallel is equal to the angle of the intersections of any other of the parallel planes with the planes not parallel.

85. If from a point without two lines are drawn to a plane, one perpendicular to the plane the other perpendicular to a given line in the plane, the straight line joining the feet of these perpendiculars is perpendicular to the given line.

86. A O, BO, CO are perpendicular to each other at the common point 0 (40); if A B is joined and O D is drawn perpendicular to A B, and C D joined, C D is also perpendicular to A B.

87. If from two points A, A', above a plane perpendiculars A B, A' B', are drawn to the plane, and a plane passed through A perpendicular to the line joining A A', its line of intersection with the given plane is perpendicular to the line joining B B.

88. If from a point in one of two intersecting planes two lines are drawn, one perpendicular to the second plane, the other perpendicular to the line of intersection of the two planes, then the plane of these two lines is perpendicular to the line of intersection of the two given planes.

89. If at the point of intersection of the perpendiculars from the vertices to the opposite sides of a triangle a perpendicular to the plane of the triangle is drawn, a line joining any vertex of the triangle to any point of the perpendicular is perpendicular to the side opposite this vertex.

90. The angle between two perpendiculars drawn from any point to two planes is equal to the angle of the planes (or to its supplement).

91. If through any point in the intersection of two planes that are perpendicular to each other two lines are drawn in one plane making equal angles with the line of intersection, then these two lines make equal angles with any line drawn in the other plane through this point.

92. If a straight line is perpendicular to a plane, its projection on any other plane will be at right angles to the intersection of the two planes.