18. To find, by a geometrical construction, an arithmetic, geometric, and harmonic mean between two given lines.

19. Prove geometrically, that an arithmetic mean between two quantities, is greater than a geometric mean. Also having given the sum of two lines, and the excess of their arithmetic above their geometric mean, find by a construction the lines themselves.

20. If through the point of bisection of the base of a triangle any line be drawn, intersecting one side of the triangle, the other produced, and a line drawn parallel to the base from the vertex, this line shall be cut harmonically.

21. If a given straight line AB be divided into any two parts in the point C, it is required to produce it, so that the whole line produced may be harmonically divided in Cand B.

22. If from a point without a circle there be drawn three straight lines, two of which touch the circle, and the other cuts it, the line which cuts the circle will be divided harmonically by the convex circumference, and the chord which joins the points of contact.

III.

23. Shew geometrically that the diagonal and side of a square are incommensurable.

24. If a straight line be divided in two given points, determine a third point, such that its distances from the extremities, may be proportional to its distances from the given points.

25. Determine two straight lines, such that the sum of their squares may equal a given square, and their rectangle equal a given rectangle.

26. Draw a straight line such that the perpendiculars let fall from any point in it on two given lines may be in a given ratio.

27. If diverging lines cut a straight line, so that the whole is to one extreme, as the other extreme is to the middle part, they will intersect every other intercepted line in the same ratio.

28. It is required to cut off a part of a given line so that the part cut off may be a mean proportional between the remainder and another given line.

29. It is required to divide a given finite straight line into two parts, the squares of which shall have a given ratio to each other.

IV.

30. From the vertex of a triangle to the base, to draw a straight line which shall be an arithmetic mean between the sides containing the vertical angle.

31. From the obtuse angle of a triangle, it is required to draw a line to the base, which shall be a mean proportional between the segments of the base. How many answers does this question admit of?

32. To draw a line from the vertex of a triangle to the base, which shall be a mean proportional between the whole base and one segment. 33. If the perpendicular in a right-angled triangle divide the hypotenuse in extreme and mean ratio, the less side is equal to the alternate segment.

34. From the vertex of any triangle ABC, draw a straight line meeting the base produced in D, so that the rectangle DB. DČ= AD2. 35. To find a point P in the base BC of a triangle produced, so that PD being drawn parallel to AC, and meeting A B produced to D, AC: CP:: CP: PD.

36. If the triangle ABC has the angle at C a right angle, and from Ca perpendicular be dropped on the opposite side intersecting it in D, then AD: DB:: AC2: CB2.

37. In any right-angled triangle, one side is to the other, as the excess of the hypotenuse above the second, to the line cut off from the first between the right angle and the line bisecting the opposite angle.

38. If on the two sides of a right-angled triangle squares be described, the lines joining the acute angles of the triangle and the opposite angles of the squares, will cut off equal segments from the sides; and each of these equal segments will be a mean proportional between the remaining segments.

39. In any right-angled triangle ABC, (whose hypotenuse is AB) bisect the angle A by AD meeting CB in D, and prove that

2AC: AC-CD2 :: BC: CD.

40. On two given straight lines similar triangles are described. Required to find a third, on which, if a triangle similar to them be described, its area shall equal the difference of their areas.

41. In the triangle ABC, AC-2.BC. If CD, CE respectively bisect the angle C, and the exterior angle formed by producing AC; prove that the triangles CBD, ACD, ABC, CDE, have their areas as 1, 2, 3, 4.

V.

42. It is required to bisect any triangle (1) by a line drawn parallel, (2) by a line drawn perpendicular, to the base.

43. To divide a given triangle into two parts, having a given ratio to one another, by a straight line drawn parallel to one of its sides. 44. Find three points in the sides of a triangle, such that, they being joined, the triangle shall be divided into four equal triangles. 45. From a given point in the side of a triangle, to draw lines to the sides which shall divide the triangle into any number of equal parts. 46. Any two triangles being given, to draw a straight line parallel to a side of the greater, which shall cut off a triangle equal to the less. VI.

47. The rectangle contained by two lines is a mean proportional between their squares.

48. Describe a rectangular parallelogram which shall be equal to a given square, and have its sides in a given ratio.

49. If from any two points within or without a parallelogram, straight lines be drawn perpendicular to each of two adjacent sides and intersecting each other, they form a parallelogram similar to the former.

50. It is required to cut off from a rectangle a similar rectangle which shall be any required part of it.

51. If from one angle A of a parallelogram a straight line be drawn cutting the diagonal in E and the sides in P, Q, shew that

AEPE. EQ.

52. The diagonals of a trapezium, two of whose sides are parallel, cut one another in the same ratio.

VII.

53. In a given circle place a straight line parallel to a given straight line, and having a given ratio to it; the ratio not being greater than that of the diameter to the given line in the circle.

54. In a given circle place a straight line, cutting two radii which are perpendicular to each other, in such a manner, that the line itself may be trisected.

55. AB is a diameter, and P any point in the circumference of a circle; AP and BP are joined and produced if necessary; if from any point C of AB, a perpendicular be drawn to AB meeting AP and BP in points D and E respectively, and the circumference of the circle in a point F, shew that CD is a third proportional of CE and CF.

56. If from the extremity of a diameter of a circle tangents be drawn, any other tangent to the circle terminated by them is so divided at its point of contact, that the radius of the circle is a mean proportional between its segments.

57. From a given point without a circle, it is required to draw a straight line to the concave circumference, which shall be divided in a given ratio at the point where it intersects the convex circumference.

58. From what point in a circle must a tangent be drawn, so that a perpendicular on it from a given point in the circumference may be cut by the circle in a given ratio?

59. Through a given point within a given circle, to draw a straight line such that the parts of it intercepted between that point and the circumference, may have a given ratio.

60. Let the two diameters AB, CD, of the circle ADBC be at right angles to each other, draw any chord EF, join CE, CF, meeting AB in G and H; prove that the triangles CGH and CEF are similar.

61. A circle, a straight line, and a point being given in position, required a point in the line, such that a line drawn from it to the given point may be equal to a line drawn from it touching the circle. What must be the relation among the data, that the problem may become porismatic, i. e. admit of innumerable solutions ?

VIII.

62. Prove that there may be two, but not more than two, similar triangles in the same segment of a circle.

63. If as in Euclid VI. 3, the vertical angle BAC of the triangle BAC be bisected by AD, and BA be produced to meet CE drawn parallel to AD in E; shew that AD will be a tangent to the circle described about the triangle EAC.

64. If a triangle be inscribed in a circle, and from its vertex, lines be drawn parallel to the tangents at the extremities of its base, they will cut off similar triangles.

65. If from any point in the circumference of a circle perpendiculars be drawn to the sides, or sides produced, of an inscribed triangle; shew that the three points of intersection will be in the same straight line.

66. If through the middle point of any chord of a circle, two chords be drawn, the lines joining their extremities shall intersect the first chord at equal distances from its extremities.

67. If a straight line be divided into any two parts, to find the locus of the point in which these parts subtend equal angles.

68. If the line bisecting the vertical angle of a triangle be divided into parts which are to one another as the base to the sum of the sides, the point of division is the center of the inscribed circle.

69. The rectangle contained by the sides of any triangle is to the rectangle by the radii of the inscribed and circumscribed circles, as twice the perimeter is to the base.

70. Shew that the locus of the vertices of all the triangles constructed upon a given base, and having their sides in a given ratio, is a circle. 71. If from the extremities of the base of a triangle, perpendiculars be let fall on the opposite sides, and likewise straight lines drawn to bisect the same, the intersection of the perpendiculars, that of the bisecting lines, and the center of the circumscribing circle, will be in the same straight line.

IX.

72. If a tangent to two circles be drawn cutting the straight line which joins their centers, the chords are parallel which join the points of contact, and the points where the line through the centers cuts the circumferences.

73. If through the vertex, and the extremities of the base of a triangle, two circles be described, intersecting one another in the base or its continuation, their diameters are proportional to the sides of the triangle.

74. If two circles touch each other externally and also touch a straight line, the part of the line between the points of contact is a mean proportional between the diameters of the circles.

75. If from the centers of each of two circles exterior to one another, tangents be drawn to the other circles, so as to cut one another, the rectangles of the segments are equal.

76. If a circle be inscribed in a right-angled triangle and another be described touching the side opposite to the right angle and the produced parts of the other sides, shew that the rectangle under the radii is equal to the triangle, and the sum of the radii equal to the sum of the sides which contain the right angle.

77. If a perpendicular be drawn from the right angle to the hypotenuse of a right-angled triangle, and circles be inscribed within the two smaller triangles into which the given triangle is divided, their diameters will be to each other as the sides containing the right angle.

X.

78. Describe a circle passing through two given points and touching a given circle.

79. Describe a circle which shall pass through a given point and touch a given straight line and a given circle.

80. Through a given point draw a circle touching two given circles.

81. Describe a circle to touch two given right lines and such that a tangent drawn to it from a given point, may be equal to a given line. 82. Describe a circle which shall have its center in a given line, and shall touch a circle and a straight line given in position.

XI.

83. Given the perimeter of a right-angled triangle, it is required to construct it, (1) If the sides are in arithmetical progression. (2) If the sides are in geometrical progression.

84. Given the vertical angle, the perpendicular drawn from it to the base, and the ratio of the segments of the base made by it, to construct the triangle.

85. Apply (vi. c.) to construct a triangle; having given the vertical angle, the radius of the inscribed circle, and the rectangle contained by the straight lines drawn from the center of the circle to the angles at the base.

86. Describe a triangle with a given vertical angle, so that the line which bisects the base shall be equal to a given line, and the angle which the bisecting line makes with the base shall be equal to a given angle.

87. Given the base, the ratio of the sides containing the vertical angle, and the distance of the vertex from a given point in the base; to construct the triangle.

88. Given the vertical angle and the base of a triangle, and also a line drawn from either of the angles, cutting the opposite side in a given ratio, to construct the triangle.

89. Upon the given base AB construct a triangle having its sides in a given ratio and its vertex situated in the given indefinite line CD. 90. Describe an equilateral triangle equal to a given triangle. 91. Given the hypotenuse of a right-angled triangle, and the side of an inscribed square. Required the two sides of the triangle.

92. To make a triangle, which shall be equal to a given triangle, and have two of its sides equal to two given straight lines; and shew that if the rectangle contained by the two straight lines be less than twice the given triangle, the problem is impossible.

XII.

93. Given the sides of a quadrilateral figure inscribed in a circle, to find the ratio of its diagonals.

94. The diagonals AC, BD, of a trapezium inscribed in a circle, cut each other at right angles in the point E;

the rectangle AB.BC: the rectangle AD.DC:: BE: ED.

XIII.

95. In any triangle, inscribe a triangle similar to a given triangle. 96. Of the two squares which can be inscribed in a right-angled triangle, which is the greater?

97. From the vertex of an isosceles triangle two straight lines