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. 1.

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:

(a) A circle has only one centre.

(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.

THEOREM I.

Circles of equal radii are identically equal.

Part. En. Let A and B be circles of equal radii; it is required to prove that they are identically equal.

Proof. Let their centres be C and D. Place the circle B upon the circle A so that the point D falls upon the point C, and take any point E outside both circles and

E

join CE. Then since all radii of the same circle are equal, and the circles are of equal radii;

therefore the distances from C along CE to the circumferences of the two circles are the same;

therefore the circumferences cut the line CE in the same point.

Similarly they cut every line through C in the same point, and therefore coincide altogether, and the two circles are identically equal.

COR. Two (different) circles whose circumferences meet one another cannot be concentric*.

*Euclid, III. 5.