44. AC one of the sides of a triangle ABC is bisected în 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.

45. Produce a given line AB to P so that AP. BP = AB2.

46. 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.

47. Given a polygon of n sides to construct an equal polygon of (n-1) sides. Hence construct a rectangle equal to any given rectilineal figure.

48. Prove that the squares on the diagonals of any parallelogram are together equal to the squares on its sides. 49. O is the point of intersection of the diagonals of a square ABCD, and P any other point whatever. Prove AP2 + BP2 + CP2 + DP2 = 40A2 + 4OP2.

that

50.

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

51. 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 hypotenuse.

52. Find the locus of the middle point of a line drawn from a given point to meet a given line.

53. 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.

54. AB is divided into two unequal parts in Cand 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.

55. 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.

56. 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.

57. 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.

A medium of a triangle is the line drawn from an angle
to the point of bisection of the opposite side.

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

59. 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.

60. 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.

61. 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.

W.

62. 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, and bisect one another at that point.

63. 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 III.

THE CIRCLE.

SECTION I.

ELEMENTARY PROPERTIES.

A circle is a plane figure contained by one line, which is called the circumference, and is such that all the lines drawn from a certain point within the figure to the circumference are equal to one another. point is called the centre of the circle.

This

A straight line drawn to the circumference from the centre is called a radius of the circle.

A straight line drawn through the centre and terminated both ways by the circumference is called a diameter of the circle.

Def. I. An arc is a part of a circumference. Def. 2. A chord of a circle is the straight line joining any two points on the circumference. When the arcs into which the chord divides the circumference are unequal, they are called the major and minor arcs respectively. Such arcs are said to be conjugate to one another.

B

Def. 3. A segment of a circle is the figure contained by a chord and either of the arcs into which the chord divides the circumference. The segments are called major and minor segments according as the arcs that bound them are major or minor arcs.

Def. 4. The conjugate angles formed at the centre of a circle by two radii are said to stand upon the conjugate arcs opposite them intercepted by the radii, the major angle upon the major arc, and the minor angle upon the minor arc.

Def. 5. A sector is the figure contained by an arc and the radii drawn to its extremities. The angle of the sector is the angle at the centre which stands upon the arc of the

sector.

Def. 6. Circles that have a common centre are said to be concentric.

The following properties of the circle are immediate consequences of Book I. Def. 8:

(b) A point is within, on, or without the circumference of a circle, according as its distance from the centre is less than, equal to, or greater than the radius.

(c) The distance of a point from the centre of a circle is less than, equal to, or greater than the radius, according as the point is within, on, or without the circumference.