21. Three given straight lines are in the same straight line ; find a point from which lines drawn to their extremities shall contain equal angles.

22. Draw through a given point in the diameter of a circle a chord, which shall form with the lines joining its extremities with either extremity of the diameter, the greatest possible triangle.

23. ADB, ACB, are the arcs of two equal circles cutting one another in the straight line AB, draw the chord ACD cutting the inner circumference in C and the outer in D, such that AD and DB together may be double of AC and CB together.

24. From a given point without a given circle a line is drawn cutting the circle. It is required to draw from the same point another line also cutting the circle, so that the sum of the arcs intercepted between these two lines shall be equal to a given arc.

25. A given straight line being divided in a given point, to find a point at which each segment of the given straight line shall subtend an angle equal to half a right angle.

26. Divide a circle into two parts such that the angle contained in one segment shall equal twice the angle contained in the other.

27. Any segment of a circle being described on the base of a triangle; to describe on the other sides segments similar to that on the base.

28. Through a given point within or without a circle, it is required to draw a straight line cutting off a segment containing a given angle.

29. Through two given points to describe a circle bisecting the circumference of a given circle.

30. A segment of a circle being described upon AB, it is required to draw a chord AC, such that CK being drawn perpendicular to AB, AC+ CK shall be a maximum.

31. In the circumference of a given circle, to determine a point to which two straight lines drawn to two given points shall contain an angle equal to a given angle, pointing out the limitations within which the problem is possible.

32. One side of a trapezium capable of being inscribed in a given circle is given, the sum of the remaining three sides is given; and also one of the angles opposite to the given side: construct it.

33. To find a point P, so that tangents drawn from it to the outsides of two equal circles which touch each other, may contain an angle equal to a given angle.

34. Given two circles: it is required to find a point from which tangents may be drawn to each, equal to two given straight lines.

35. Between two given circles to place a straight line terminated by them, such that it shall equal a given straight line, and be inclined at a given angle to the straight line joining their centres.

36. Two circles being given in position and magnitude, draw a straight line cutting them, so that the chords in each circle may be equal to a given line not greater than the diameter of the smaller circle.

37. Describe two circles with given radii which shall cut each other, and have the line between the points of section equal to a given line.

こ 38. If two circles cut each other; to draw from one of the points of intersection a straight line meeting the circles, so that the part of

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it intercepted between the circumferences may be equal to a given line.

39. Three points being in the same plane, find a fourth, where lines drawn from the former three shall make given angles with one another.

40. Two given circles touch each other internally. Find the semichord drawn perpendicularly to the diameter passing through the point of contact, which shall be bisected by the circumference of the inner circle.

41. The circumference of one circle is wholly within that of another. Find the greatest and the least straight lines that can be drawn touching the former and terminated by the latter.

42. Draw a straight line through two concentric circles, so that the chord terminated by the exterior circumference may be double that terminated by the interior. What is the least value of the radius of the interior circle for which the problem is possible?

43. To draw a straight line cutting two concentric circles, so that the part of it which is intercepted by the circumference of the greater may be triple the part intercepted by the circumference of the less.

44. Find the greatest of all triangles having the same vertical angle and equal distances between that angle and the bisection of the opposite sides.

45. If a string of a given length be fixed at each end to twò given points A and B, and be pulled downwards, so as to form a triangle with the line joining A and B, determine the lowest point that it may be made to reach.

46. From a given point without a circle, at a distance from the circumference of the circle not greater than its diameter, draw a straight line to the concave circumference which shall be bisected by the convex circumference.

47. Find a point in the circumference of a circle, from whence a line drawn, making a given angle with a given radius, may be equal to a given straight line.

48. To find within an acute-angled triangle a point from which, if straight lines be drawn to the three angles of the triangle, they shall make equal angles with each other.

49. From two lines, including a given angle, cut off by a line of given length, a triangle equal to a given rectilineal figure.

50. From the extremities of the diameter of a given semicircle, draw two chords to meet in the circumference, which shall intercept a given length on a given oblique chord.

51. The positions of three stations, A, B, and C, have been laid down on a map, and an observer at D (a station in the same horizontal plane as A, B, and C,) determines the angles ADB and BDC; give a geometrical construction for laying down D on the map.

52. In an acute-angled triangle, to find a point from which if three lines be drawn to the three angles, the sum of these lines shall be the least possible.

53. Draw lines from the angles of a triangle to the points of bisection of the opposite sides and terminated in these points. If from the extremities of any one of them, lines be drawn parallel to the remaining two and produced to meet, a triangle will be formed whose sides are equal to the three lines first drawn.

54. It it required within an isosceles triangle to find a point such,

that its distance from one of the equal angles may be double its distance from the vertical angle.

55. To construct an isosceles triangle equal to a scalene triangle and having an equal vertical angle with it.

56. Given the angle at the base of an isosceles triangle, and the perpendicular from it on the opposite side, to construct the triangle. 57. Given the base, the vertical angle, and the differences of the sides, to construct the triangle.

58. 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.

59. Given the perpendicular height, the vertical angle and the sum of the sides, to construct the triangle.

60. Construct a triangle in which the vertical angle and the difference of the two angles at the base shall be respectively equal to two given angles, and whose base shall be equal to a given straight line.

61. Given the vertical angle, the difference of the two sides containing it, and the difference of the segments of the base made by a perpendicular from the vertex; construct the triangle.

62. On a given straight line to describe a triangle having its vertical angle equal to a given angle, and the difference of its sides equal to a given line.

63. Given the vertical angle, and the lengths of two lines drawn from the extremities of the base to the points of bisection of the sides, to construct the triangle.

64. Given the base, and vertical angle, to find the triangle whose area is a maximum.

65. Find a triangle of which the vertical angle, the sum of the squares of the two sides containing it and the area are given.

66. The base, vertical angle, and rectangle under the sum of the other sides and one of them are given. Construct the triangle.

67. Describe a circle the circumference of which shall pass through a given point and touch a given circle in a given point. 68. Describe a circle which shall pass through a given point and which shall touch a given straight line in a given point.

69. Describe a circle to touch two right lines given in position and such that a tangent drawn to it from a given point may be equal to a given line.

70. Let AB, AC be any two lines given in position; DE a line of given length; find the position of that circle which is touched by both the lines AB, AC and whose diameter is equal to DE.

71. Describe a circle to touch two right lines given in position, so that lines drawn from a given point to the points of contact shall contain a given angle.

72. Describe a circle through a given point, and touching a given straight line, so that the chord joining the given point and point of contact may cut off a segment containing a given angle.

73. To describe a circle through two given points to cut a straight line given in position, so that a diameter of the circle drawn through the point of intersection shall make a given angle with the line.

74. The straight lines drawn from the same point, and touching the same circle, are equal. Having proved this, exhibit a construction

that shall include all the triangles which can be described with a given perimeter and given vertical angle.

75. A flag-staff of a given height is erected on a tower whose height is also given: at what point on the horizon will the flag-staff appear under the greatest possible angle.

76. Find a point from which, if straight lines be drawn to touch three given circles, none of which lies within the other, the tangents so drawn shall be equal.

77. The centres of three circles (A, B, C,) are in the same straight line, B and C touch each other externally and A internally, if a line be drawn through the point of contact of B and C, making any angle with the common diameter, then the portion of this line intercepted between C and A, is equal to the portion intercepted between B and A.

78. If P be a point without a circle whose centre is O, and AOB a diameter perpendicular to PO: draw a line PMEC cutting the circle in M and C and the diameter in E, so that the rectangle PM, PC, may be four times the rectangle AE, EB.

79. From a given point without a circle draw a straight line cutting the circle, so that the rectangle contained by the part of it without and the part within the circle shall be equal to a given square.

80. Let AP be a tangent to any circle, and AB a diameter. To determine the point P, so that PCB being drawn, cutting the circumference in C, the rectangle contained by PC, CB, shall be equal to a given square; and shew in what cases this is impossible.

81. The diameter ACD of a circle, whose centre is C, is produced to P, determine a point F in the line AP such that the rectangle PF. PC may be equal to the rectangle PD. PA.

82. Through a given point draw a line terminating in two lines given in position, so that the rectangle contained by the two parts may be equal to a given rectangle.

83. A ladder is gradually raised against a wall; find the locus of its middle point.

84. A, B, C, D are four points in order in a straight line, find a point E between B and C, such that AE. EB=ED. EC by a geometrical construction.

85. Determine the locus of the extremities of any number of straight lines drawn from a given point, so that the rectangle contained by each, and a segment cut off from each by a line given in position, may be equal to a given rectangle.

86. Find a point without a given circle from which if two tangents be drawn to it, they shall contain an angle equal to a given angle, and shew that the locus of this point is a circle concentric with the given circle.

87. Find the locus of the vertex of a triangle described on a given base; (1) when the sum of the angles at the base is given; (2) when one of them is always double of the other.

88. Find the locus of the centres of all circles which cut off from the directions of two sides of a triangle, chords equal to two given straight lines.

Hence describe a circle that shall cut off from the direction of three sides of a triangle, chords respectively equal to three given straight

89. In a given straight line to find a point at which two other straight lines, being drawn to two given points, shall contain a right angle. Shew that if the distance between the two given points be greater than the sum of their distances from the given line, there will be two such points; if equal, there may be only one; if less, the problem may be impossible.

90. Produce a given straight line so that the rectangle under the given line, and the whole line produced may equal the square of the part produced.

91. To produce a given straight line, so that the rectangle contained by the whole line thus produced, and the part of it produced, shall be equal to a given square.

92. To determine a right-angled triangle whose hypothenuse may be equal to a given straight line, and the rectangle contained by whose sides may be equal to the square of their difference.

93. Given the lengths of the three lines drawn from the angles of a triangle to the points of bisection of the opposite sides, construct the triangle.

94. Describe a triangle whose sides shall be bisected by three given straight lines, and one of whose sides shall pass through a given point.

95. Find the locus of a point, such that if straight lines be drawn from it to the four corners of a given square, the sum of the squares shall be invariable.

THEOREMS.

5. The arcs intercepted between any two parallel chords in a circle are equal. Also the sum of the arcs subtending the vertical angle made by any two chords that intersect, is the same, as long as the angle of intersection remains the same.

6. From the extremities A and C of a given circular arc AC, equal arcs AB, CD are measured in opposite directions: prove that the chords AC, BD are parallel.

7. Two circles cut each other, and from the points of intersection straight lines are drawn parallel to one another, the portions intercepted by the circumferences are equal.

8. A, B, C, A', B', C' are points on the circumference of a circle; if the lines AB, AC be respectively parallel to A'B', A'C', shew that BC' is parallel to B'C.

9. C, C' are the centres of two circles of unequal radii, CR, C'R' any pair of parallel radii, join RR': then shall all such lines produced meet in one point. Prove this property, and from it deduce à method of drawing a common tangent to two circles.

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10. If from a given point a straight line be drawn touching a circle given in position, the straight line is given in position and magnitude.

11. DF is a straight line touching a circle, and terminated by AD, BF, the tangents at the extremities of the diameter AB, shew that the angle which DF subtends at the centre is a right angle.