51. If from any point in the arc of a sector perpendiculars be let fall on the bounding radii, the distance between their feet will be constant.

52. If on two straight lines containing an angle, segments of circles be described containing angles equal to it, and falling without it, the lines produced will touch the segments.

53. From a given point without a circle, a straight line is drawn cutting the circle. Draw from the same point another line so as to intercept two arcs which together shall subtend at the centre an angle equal to a given angle.

54. If a circle be described about a triangle ABC, and perpendiculars be let fall from A, B, C on the opposite sides and produced to meet the circle in D, E, F, then will the arcs EF, FD, DE be bisected in A, B, C.

55. Given the vertical angle of a triangle, the sum of its arms, and also the difference of its arms; construct the triangle.

56. Find a point within a triangle such that, if straight lines be drawn from it to the three angles of a triangle, they shall make equal angles with one another. problem always possible?

Is this

57. If two straight lines cut at right angles in a circle, the squares on the four segments are together equal to the square on the diameter.

58. If from two given points in the diameter of a circle, equidistant from the centre, straight lines be drawn. to any point in the circumference the sum of their squares is constant.

59. If from any point without a circle lines be drawn cutting the circle and making equal angles with the longest line, they will cut off equal segments.

60. If from the angles of an equilateral triangle, inscribed in a circle, straight lines are drawn to any point in the circumference, then shall the two shorter lines thus drawn be equal to the longer one.

61. If any point be taken in the arc of a segment of a circle, cut off by one of the sides of an equilateral triangle inscribed in the circle, and straight lines drawn from it to the extremities of the base of the segment, then the sum of the squares on them together with the rectangle contained by them, shall be constant.

62. If from the angles of an equilateral triangle inscribed in a circle, straight lines are drawn to any point in the circumference, then shall the sum of the squares described upon them be constant.

63. The locus of the point, the sum of the squares of whose distances from the angular points of a given equilateral triangle is equal to twice the square on one of the sides, is the circumscribed circle.

64. Through a given point situated between two lines inclined at a given angle, to draw a line, having its extremities terminated in these lines such that the rectangle contained by the parts may be the least possible.

65. Draw lines from two given points meeting in a given straight line, and containing the greatest possible angle.

66. AB is the diameter of a circle, MN a chord parallel to AB, in AB take any point P, join PM, PN,

and prove that the squares on PM, PN are together equal to the squares on AP, PB.

67. AB is the diameter of a circle, and tangents are drawn at the extremities A, B meeting a tangent at any point in D, F, prove that the angle subtended by FD at the centre is a right angle.

68. From a given point without a circle draw a straight line cutting the circle such that the intercepted chord shall be equal to the part without the circle.

69. Find the locus of the middle point of a rod which is moved so as always to have its extremities in the arms of a given right angle.

70. Describe an isosceles triangle having the angle at the vertex four times as large as each of the angles at the base.

71. Describe an isosceles triangle having the angle at the vertex treble each of the angles at the base.

72. Describe a right-angled triangle having one of the acute angles one-fourth of the other.

73. Describe a right-angled triangle having one of the acute angles half as large again as the other.

74. Describe a triangle having one of its angles half of another and treble the remaining angle.

I.

BOOK IV.

Describe a circle touching one side of a triangle, and the other two sides produced.

2.

On a given straight line describe a regular hexagon.

3. Find a point which is equally distant from three given points. Is this problem always possible?

4. Inscribe a circle in a rhombus.

Can a circle be described about a rhombus ?

5. Describe a parallelogram about a circle, having an angle equal to a given angle.

6. If a parallelogram be described about a circle, it will be equilateral.

7. Determine the condition that the straight line joining the centres of the circles inscribed in, and described about a given triangle, may pass through one of the angular points of the triangle.

8. What relation exists between the sides of a triangle when the centres of the circles inscribed in and described about the triangle coincide?

9. Describe a regular hexagon having its angular points in the sides of a given equilateral triangle.

IO. If the chords which bisect two angles of a triangle inscribed in a circle be equal, prove that either the angles are equal or else the third angle is equal to the angle of an equilateral triangle.

II. If one square be inscribed in another, the difference between their areas is equal to twice the rectangle contained by the segments of any one of the sides of the larger square.

12. If one square be inscribed in another, the area of the smaller square is equal to the squares on the segments of a side of the larger square.

13. Apply the previous deduction to shew how the square described on the hypothenuse of a right-angled triangle may be divided into parts which, when properly arranged, will form the squares on the sides containing the right angle.

14. If a quadrilateral figure be circumscribed about a circle, then two of its opposite sides are together equal to the other two.

15. If two of the opposite sides of a quadrilateral figure are equal to the other two, then a circle can be inscribed within it.

16. If a quadrilateral figure be circumscribed about a circle, then will the angles subtended at the centre by two of its opposite sides be together equal to two right angles. Three circles touch each other externally, prove that their centres are the angular points of the triangle described about the circle which passes through the points of contact.

18. The circles each of which touches two sides of a regular pentagon at the extremity of a third meet in a