isosceles triangle, and the opposite side (produced if necessary) be equal to a side of the triangle, the angle formed by this line and the base produced, is equal to three times either of the equal angles of the triangle.

27. In the base BC of an isosceles triangle ABC, take a point D, and in CA, take CE equal to CD, let ED produced meet AB produced in F; then 3.AEF = 2 right angles + AFE, or = 4 right angles + AFE.

28. If from the base to the opposite sides of an isosceles triangle, three straight lines be drawn, making equal angles with the base, viz. one from its extremity, the other two from any other point in it, these two shall be together equal to the first.

29. A straight line is drawn, terminated by one of the sides of an isosceles triangle, and by the other side produced, and bisected by the base; prove that the straight lines, thus intercepted between the vertex of the isosceles triangle, and this straight line, are together equal to the two equal sides of the triangle.

30. In a triangle, if the lines bisecting the angles at the base be equal, the triangle is isosceles, and the angle contained by the bisecting lines is equal to an exterior angle at the base of the triangle.

31. No two straight lines drawn from the angles of a triangle and terminated by the opposite sides can bisect each other.

32. In a triangle, if lines be equal when drawn from the extremities of the base, (1) perpendicular to the sides, (2), bisecting the sides, (3) making equal angles with the sides; the triangle is isosceles : and these lines which respectively join the intersections of the sides, are parallel to the base.

33. If one angle of a triangle be double or triple of another; or if it be equal to the sum, or to one half of the difference of the remaining two; the triangle may either be divided into two isosceles triangles, or else an isosceles triangle may be added to it in such a manner as to form together with it a single isosceles triangle. Prove this, and distinguish the different cases.

II.

34. If two straight lines are respectively at right angles to two others which intersect, shew that each pair of lines includes the same angle.

35. ABC is a triangle right-angled at B, and having the angle A double the angle C; shew that the side BC is less than double the side AB.

36. If one angle of a triangle be equal to the sum of the other two, the greatest side is double of the distance of its middle point from the opposite angle.

37. If from the right angle of a right-angled triangle, two straight lines be drawn, one perpendicular to the base, and the other bisecting it, they will contain an angle equal to the difference of the two acute angles of the triangle.

38. If the vertical angle CAB of a triangle ABC be bisected by AD, to which the perpendiculars CE, BF are drawn from the remaining angles: bisect the base BC in G, join GE, GF, and prove these lines equal to each other.

39. The difference of the angles at the base of any triangle, is double of the angle contained by a line drawn from the vertex perpendicular to the base, and another bisecting the angle at the vertex.

40. If one angle at the base of a triangle be double of the other, the less side is equal to the sum or difference of the segments of the base, made by the perpendicular from the vertex, according as the angle is greater or less than a right angle.

41. If two exterior angles of a triangle be bisected, and from the point of intersection of the bisecting lines, a line be drawn to the opposite angle of the triangle, it will bisect that angle.

42. From the vertex of a scalene triangle draw a right line to the base, which shall exceed the less side as much as it is exceeded by the greater.

43. Divide into three equal angles, (1) a right angle; (2) onefourth of a right angle.

44. Prove that the sum of the distances of any point within a triangle from the three angles is greater than half the perimeter of the triangle.

45. If from the angles of a triangle ABC, straight lines ADE, BDF, CDG be drawn through a point D to the opposite sides, prove that the sides of the triangle are greater than the three lines drawn to the point D, and less than twice the same, but greater than two-thirds of the lines drawn through the point to the opposite sides.

46. If two triangles have the same base as a third triangle, and their vertices upon its two sides, the sum of their sides will be less than twice the sum of the sides of the third triangle, and the sum of their vertical angles will be greater than twice its vertical angle.

47. In a plane triangle an angle is right, acute or obtuse, according as the line joining the vertex of the angle with the middle point of the opposite side is equal to, greater or less than half of that side.

48. If the straight line AD bisect the angle A of the triangle ABC, and BDE be drawn perpendicular to AD and meeting AC or AC produced in E, shew that BD = DE.

49. The side BC of a triangle ABC is produced to a point D. The angle ACB is bisected by a line CE which meets AB in E. A line is drawn through E parallel to BC and meeting AC in F, and the line bisecting the exterior angle ACD, in G. Then EF is equal to FG.

50. The sides AB, AC, of a triangle are bisected in D and E respectively, and BE, CD, are produced until EF = EB, and GD=DC; shew that the line GF passes through A.

51. AEB, CED are two straight lines intersecting in E; lines AC, DB are drawn forming two triangles ACE, BED; if the angles ACE, DBE be bisected by the straight lines CF, BF meeting in F, shew that the angle CFB is equal to half the sum of the angles EAC, EDB.

52. In a triangle ABC, AD being drawn perpendicular to the straight line BD which bisects the angle B, shew that a line drawn from D parallel to BC will bisect AC.

53. If the sides of a triangle be trisected and lines be drawn through the points of section adjacent to each angle so as to form another triangle, this shall be in all respects equal to the first triangle.

54. Between two given straight lines it is required to draw a straight line which shall be equal to one given straight line, and parallel to another.

55. ABC is a given triangle, it is required to draw from a given point P, in the side AB, or AB produced, a straight line to AC, so that it shall be bisected by BC.

56. If from the vertical angle of a triangle three straight lines be drawn, one bisecting the angle, another bisecting the base, and the third perpendicular to the base, the first is always intermediate in magnitude and position to the other two.

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

58. In the base of a triangle, to find a point from which if two lines be drawn, (1) perpendicular, (2) parallel, to the two sides of the triangle, their sum shall be equal to a given line.

III.

59. In the figure of Euc. i. 1, the given line is produced to meet either of the circles in P; shew that P and the points of intersection of the circles, are the angular points of an equilateral triangle.

60. If each of the equal angles of an isosceles triangle be onefourth of the third angle, and from one of them a line be drawn at right angles to the base meeting the opposite side produced; then will the part produced, the perpendicular, and the remaining side, form an equilateral triangle.

61. In the figure Euc. I. 1, if the sides CA, CB of the equilateral triangle ABC be produced to meet the circles in F, G, respectively, and if C' be the point in which the circles cut one another, on the 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.

62. 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.

63. 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.

64. 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.

65. Describe a right-angled triangle upon a given base, having given also the perpendicular from the right angle upon the hypotenuse. 66. 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.

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

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

69. 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.

70. Make an isosceles triangle of given altitude whose sides shall pass through two given points, and whose base shall be on a given straight line.

71. Given of any triangle the perpendicular let fall from the ver

tical angle on the base, and the difference between each segment made by the perpendicular and its adjacent side, construct the triangle.

72. Having given the straight lines which bisect the angles at the base of an equilateral triangle, determine a side of the triangle.

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

74. 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. 75. Given the perimeter and the angles of a triangle; to construct it. 76. 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.

77. 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.

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

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

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

82. 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.

83. Construct a triangle two of whose sides shall be together: double of the third side. If the triangle be isosceles, prove that it is also equilateral.

84. In a right-angled triangle, given the sums of the base and hypotenuse, and of the base and perpendicular, also of the perpendicular and hypotenuse; to construct the triangle.

85. Given the base, the difference and the sum of the other two sides; to construct the triangle.

86. Construct a triangle equiangular to a given triangle, and having its angular points upon three given straight lines which meet in a point.

87. If through the angular points of any triangle straight lines be drawn, making equal angles with the sides taken in order, the triangle formed by these lines shall be equiangular to the original triangle.

88. Given the base of a triangle, one of the angles at the base, and also the angle which the perpendicular drawn upon the base from the opposite angle makes with the side opposite to the given angle of the triangle; construct the triangle.

89. With the three altitudes of a triangle ABC as sides, form a triangle; and with the three altitudes of the triangle so formed, form another triangle: shew that this third triangle is similar to the original triangle ABC. Hence determine a triangle having given its three altitudes.

V.

90. 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. 91. Through a given point 4, 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.

92. 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.

93. 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.

94. 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.

95. 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.

96. AD, BC are two parallel straight lines, cut obliquely by AB and perpendicularly by AC; BED is drawn cutting AC in E, so that ED is equal to twice BA; prove that the angle DBC is equal to onethird of the angle ABC.

97. Having given the angles and diagonals of a parallelogram; construct it.

98. The four straight lines which bisect the angles of a parallelogram either meet in a point or form a parallelogram.

Find a point such that the perpendiculars let fall from it upon two given straight lines shall be respectively equal to two given straight lines. How many such points are there?

100. Draw a line parallel to one of the sides of a triangle, such that the portion of it intercepted between the other two sides, shall be equal to the difference between one of those sides, and the side parallel to the line.

101. If ABC be a triangle in which C is a right angle, draw`a straight line parallel to a given straight line so as to be terminated by CA, CB and bisected by AB.

102. On the sides AB, BC, CD of a parallelogram are described the equilateral triangles ABE, CDE without, and BCG within the figure; prove that EG is equal to one, and FG to the other diagonal.

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

104. 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.

105. ABCD, A'B'C'D' are two parallelograms whose corresponding sides are equal, but the angle 4 is greater than the angle '; prove that the diameter AC is less than A'C', but BD greater than B'D.

106. 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.

107. 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 equiangular to the original one.

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