64. The distance between two parallels is a; any line AB is drawn between the parallels, and the two supplement

ary angles A and B are bisected; how great is the altitude of the right-angled triangle, formed thereby? 65. In a right-angled triangle, the right angle is bisected, from the point where the bisecting line cuts the hypothenuse, lines are drawn parallel to the two sides containing the right angle. Prove that the figure contained by these two lines and the two sides is a square.

66. The point C bisects the line AB. The three points are projected on to any line in A1, B1 and C1; shew that C1 bisects A,B1.

1

67. Two points A and B lie on the same side of a straight line. From A a line is drawn to the point C, which lies symmetrically with B. This line cuts the given line in a point D. Prove that AD and BD make equal angles with the given line. 68. A line is parallel to the diagonal of a parallelogram; shew that the one pair of opposite sides cut off a part of the line, equal to the part cut off by the other pair. 69. Prove that the opposite sides of a square cut off a part

of any line, equal to the part cut off a line perpendicular to this, by the other two opposite sides.

70. In a right-angled triangle squares are described on the sides containing the right angle, and from the outermost angular points of these, perpendiculars are dropped on the hypothenuse; prove that the sum of the perpendiculars equals the hypothenuse.

71. From a point in the base of an isosceles triangle, perpendiculars are drawn to both the sides. Prove that the sum of these perpendiculars equals the altitude from an extremity of the base.

72. Which parallelograms can be inscribed in a circle, and which can be circumscribed?

73. Prove that a polygon can be constructed, when all its parts are known, except three, which follow after each

other ( side and 2 angles or 2 sides and 1 angle). Which proposition on the congruence of polygons follows from this?

74. In a sector of 60° a circle is inscribed; prove that its radius is a third of the radius of the circle to which the sector belongs.

75. Which is the longest line that can be drawn between the circumferences of two circles through their one point of intersection.

76. Two given lines intersect in 0. In each of the lines any point, A and B, is chosen, and a triangle ABC with given angles is constructed. The angle at C is equal to the angle at O and lies to the same side of AB. Shew that C falls in the same line through 0, wherever A and B are chosen.

77. From a point a line is drawn to the centre of a circle, and on this line as diameter, another circle is described. Prove that this circle bisects all chords passing through the given point.

78. In a right-angled triangle a circle touches one of the sides containing the right angle and also the other side at its extremity; prove that the part which the circle cuts off from the hypothenuse is equal to twice the altitude of the triangle. 79. In a parallelogram a diagonal is drawn, and through a point in this, lines parallel to the sides of the parallelogram. The parallelogram is thereby divided into four smaller parallelograms. Prove that the two of these, through which the diagonal does not pass, are equal. 80. Prove that the one part of the altitude of a triangle, reckoned from the point of intersection of the altitudes to the foot, is equal to the prolongation of the altitude to the circumference of the circumscribed circle. 81. A triangle is divided into two other triangles by a line

from an angular point to the point of contact of the opposite side with the inscribed circle. Shew that the

circles inscribed in the smaller triangles touch the side, which the triangles have in common, at the same point. 82. Through the vertex of an angle and a given point in the line bisecting the angle any circle is described; prove that the sum of the two chords, which the circle cuts off from the legs of the angle, is constant.

83. On the three sides of a triangle arcs are described inwardly, containing angles of 120°; shew that the three arcs pass through the same point.

84. On the three sides of a triangle equilateral triangles are described outwardly. Shew that the three, lines joining the outermost angular points of the equilateral triangles with the opposite angular points of the given triangle 1) are equal, 2) cut each other at angles of 120°, and 3) pass through the same point.

85. Prove that a quadrilateral can be inscribed in a circle, when its opposite angles are supplementary.

86. On the three sides of a triangle squares are described outwardly. Prove that the three lines, joining the extremities of the outermost sides of the three squares, are twice as great as the median lines of the triangle, and perpendicular to these.

87. Shew that the altitudes of a triangle bisect the angles of another triangle, having its angular points at the feet of the altitudes. (Find in the figure systems of four points lying in the circumference of the same circle). Prove by the help of this proposition, that the three altitudes intersect in one point.

88. Any tangent is drawn to a circle with centre O. Let it cut a fixed line through O in M and mark off on the tangent MP: MO. What is the locus of P?

=

89. In a given circle a triangle is inscribed, the two angular points of which are fixed, whilst the third moves on the arc. Which are the loci of the point of intersection of the altitudes of the triangle, and of the centre of the inscribed circle?

90. AB is a fixed chord, C a moveable point of the circumference. AC is produced to D, so that CD

the locus of D.

CB. Find

91. From any point in the circumference of a circle lines are drawn to the angular points of an inscribed equilateral triangle; shew that one of the three lines equals the sum of the other two.

92. Two men, both living in the neighbourhood of a lake, jointly own a boat. Where should this be placed on the shore, so as to be equally distant from them both? 93. Construct a triangle, having given an angle, the oppo

site side, and the median line to one of the other sides. (Seek the middle point of this side, when the given side is fixed).

94. Describe a circle with given radius, passing through two given points or touching two given straight lines.

95. Construct a triangle, having given a side, the altitude and median line to this.

96. Construct a triangle, having given a side, the altitude to it, and the opposite angle.

97. Construct a quadrilateral ABCD, having given AB, BC, AC, BD and ▲ D.

98. Construct a quadrilateral ABCD, which can be inscribed in a circle, having given ▲ A, AB, AC and BD.

99. Given a line and in it the point A, and outside the line the point P. Find in the given line a point X, such that AX+XP equals a given line.

100. In a given line determine a point, such that lines from it to two given points on the same side of the line make equal angles with it. (Ex. 67).

101. In a given sector to inscribe a circle.

102. In a given circle to draw a chord equal and parallel to a given line.

103. Construct a right-angled triangle, having given a point in each of the sides containing the right angle, two points in the hypothenuse, and the length of the altitude to the hypothenuse.

104. Construct a triangle, having given the centres of its escribed circles (Ex. 87).

105. Each side of a square is divided into two parts m and n, so that two equal parts nowhere meet together. Prove that the quadrilateral, having its angular points in each of the four points of division, is a square. (Turn revolution). 106. In a regular pentagon all the diagonals are drawn. Prove that a new regular pentagon is formed thereby. 107. Prove that two diagonals in a regular hexagon are par

allel, and that a third diagonal is perpendicular to the two preceding ones, and parallel to two of the sides of the hexagon, and that three of the diagonals form an equilateral triangle.

III.

I. SIMILAR FIGURES.

80. When several parallels cut off equal parts of a straight line, the parts which they cut off from any other straight lines, will also be equal.

If AB BC and if DG and EH be drawn parallel to

AB and EH

therefore

AC, we have

H