40. Find the locus of the vertices of triangles of equal area upon the same base.

41.

Find the locus of a point, such that the sum of the squares on its distances from two given points is equal to the square on the distance between the two points.

42. If m and n are any numbers, and lines be taken whose lengths are m2+n2, m3 − n2 and 2mn units respectively, shew that these lines will form a right-angled triangle. Give examples of these triangles.

43. Through two given points on opposite sides of a straight line draw two straight lines to meet in that line, so that the angle which they form shall be bisected by that line.

44. Through a given point draw a line such that the perpendiculars on it from two given points may be equal.

45. Find points D, E in the equal sides AB, AC of an isosceles triangle ABC, such that BD = DE = EC.

46. Given two points and a straight line of indefinite length, construct an equilateral triangle so that two of its sides shall pass through the given points, and the third shall be in the given straight line.

47. Construct an isosceles triangle having the angle at the vertex double of the angles at the base.

48. Bisect a triangle by a line passing through one of its angular points.

49. Bisect a triangle by a line passing through a point in one of its sides.

50. Bisect a parallelogram by a line passing through any given point.

51. Construct a triangle equal to a given quadrilateral figure.

52. Bisect a given quadrilateral figure by a line drawn from one of its angular points.

53. Bisect a given five-sided figure by a line drawn from one of its angular points.

54. Produce a given straight line to such a distance. that the square on the produced part may be double of the square on the given line.

55. Produce a given straight line to such a distance. that the square on the whole line may be double of the square on the given line.

56. Given two sides and a median, construct the triangle.

57. Divide a straight line into two parts such that the square on one part may be four times the square on the other.

58. From B, one of the angles of a triangle ABC, a perpendicular BD is let fall on AC. Shew that the difference of the squares on AB, BC is equal to the difference of the squares on AD, DC.

59. AC one of the sides of a triangle ABC is bisected in D: and BD joined. Shew that the squares on AB and BC together are equal to twice the square on BD, and twice the square on AD.

60.

AB.

61.

Produce a given line AB to P so that AP.BP=

ABCD is the diameter of two concentric circles, P, Q any points on the outer and inner circles respectively. Prove that BP2 + CP2 = AQ2 + DQ3.

62. Prove that the squares on the diagonals of a rectangle are together equal to the squares on its sides.

63. Prove that the squares on the diagonals of any parallelogram are together equal to the squares on its sides.

Prove

64. O is the point of intersection of the diagonals of a square ABCD, and P any other point whatever. AP2 + BP2 + CP2 + DP2 = 40A2 + 4OP2.

that

65. Given the base, difference of sides, and difference of angles at the base, construct the triangle.

66. If from one of the acute angles of a right-angled triangle a line be drawn to the opposite side, the squares on that side and the line so drawn are together equal to the squares on the segment adjacent to the right angle and on the hypothenuse.

67. If from the right angle C of a right-angled triangle ABC straight lines be drawn to the opposite angles of the square on AB, the difference of the squares on these two lines will equal the difference of the squares on AC and BC.

68. AB is divided into two unequal parts in C and equal parts in D; shew that the squares on AC and BC are greater than twice the rectangle AC× CB by four times the square on CD.

69. In any right-angled triangle the square on one of the sides containing the right angle is equal to the rectangle contained by the sum and difference of the other two sides.

70. In any isosceles triangle ABC, if AD is drawn from A the vertex to any point D in the base, shew that AB2 = AD2 +BD. DC.

71.

Prove that four times the sum of the squares on the medians of a triangle is equal to three times the sum of the squares on the sides of the triangle.

72. The square of the base of an isosceles triangle is double the rectangle contained by either side, and the projection on it of the base.

73. The squares on the diagonals of a quadrilateral are double of the squares on the sides of the parallelogram formed by joining the middle points of its sides.

74. Hence shew that they are also double of the squares on the lines which join the points of bisection of the opposite sides of the quadrilateral.

75. The squares on the diagonals of a quadrilateral are.together less than the squares on the four sides by four times the square on the line joining the points of bisection of the diagonals.

76. In any quadrilateral figure the lines which join the middle points of opposite sides intersect in the line which joins the middle point of the diagonals.

77. The locus of a point which moves so that the sum of the squares of its distances from three given points is constant is a circle.

BOOK II. THE CIRCLE.

INTRODUCTION.

Def. 1.

IF a point moves in a plane so that its distance from a fixed point is constant, it traces out a line which is called the circumference of a circle.

Def. 2. The fixed point is called the centre of the circle. Def. 3. The distance of any point on the circumference from the centre is called the radius.

Def. 4. A line through the centre, terminated both ways by the circumference, is called a diameter of a circle.

Def. 5. Any portion of a circumference is called an arc.

Def. 6. The figure enclosed by an arc and the radii to its extremities is called a sector.

Def. 7. The line joining the extremities of an arc is called a chord of that arc.

Def. 8.

The parts into which a chord divides a circle are called segments.

W. G.

G