EXERCISES.

The following Theorems, depending for their demonstration upon those already demonstrated, are introduced as exercises for the pupil. In some of them references are made to the propositions upon which the demonstration depends. They are not connected with the propositions in the following books, and can be omitted if thought best.

40. Every diameter bisects the circle and the circumference. 41. A straight line can meet the circumference of a circle in only two points. (4.) (I. 51.)

42. The diameter is greater than any other chord of the circle.

43. In the same or equal circles, when the sum of the arcs is less than a circumference, the greater arc is subtended by the

greater chord; and, conversely, the greater A chord is subtended by the greater arc.

Draw AC. (21.) (I. 47.)

What is the case when the sum of the arcs

is greater than a circumference?

D

B

C

44. Equal chords are equally distant from the centre; and of two unequal chords the greater is nearer the centre.

45. The shortest and the longest line that can be drawn from any point to a given circumference lies on the line that passes from the point to the centre of the circle.

46. Two parallels cutting the circumference of a circle intercept equal arcs.

47. A straight line perpendicular to a

diameter at its extremity is a tangent to the circumference.

Draw CB. (I. 51.) .

48. The lines joining the extremities of

two diameters are parallel.

B

49. If the extremities of two chords are joined, the triangles thus formed are similar.

50. If two circumferences cut each other, the chord which joins their points of intersection is bisected at right angles by the line joining their centres. (17.)

51. If two circumferences touch each other, their centres and point of contact are in the same straight line, perpendicular to the tangent at the point of contact. (47.)

52. The distance between the centres of two circles whose circumferences cut one another, is less than the sum, but greater than the difference, of their radii.

53. Every angle inscribed in a segment greater than a semicircle is acute; and every angle inscribed in a segment less than a semicircle is obtuse.

(21.)

54. The angle made by a tangent and a chord is measured by half the included arc. Draw the diameter A B. (47.) (21.)

55. The angle formed by two chords cut- A ting each other within the circle is measured by half the sum of the intercepted arcs.

Join BC (in lower figure). (21.)

56. By moving the point of intersection of the two chords, show that (14) and (21) can be deduced from (55).

57. The segments of two chords cutting each other within a circle are reciprocally proportional.

Join AD, BC.. (21.) (II. 20.)

C

E

A

B

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58. The opposite angles of a quadrilateral inscribed in a circle are supplementary. (21.)

59. A quadrilateral whose opposite angles are supplementary, and no other, can be inscribed in a circle.

60. Circles are as the squares of their radii, or diameters, or circumferences. (32.)

61. The area of a sector is equal to half the product of its arc by the radius of the circle. (31.)

ID

B

62. Show how to find the area of a segment of a circle.

63. The area of a circumscribed polygon is equal to half the product of its perimeter by the radius of the circle.

64. A tangent is a mean proportional

between a secant drawn from the same point and the part of the secant without circle.

Join AD, DC. (54; 21.) (II. 57.)

65. The angle formed by two secants, two tangents, or a secant and a tangent cutting each other without the circle, is measured by half the difference of the intercepted arcs.

Join CF. (I. 39.) (21.)

66. By moving the point of intersection, show that (21) can be deduced from (65). Show also that (46) can be deduced from (65).

67. Two secants drawn from the same point are to each other inversely as the

parts of the secants without the circle.

Join CF, DG. (21.) (II. 57.)

A
Λ

B

D

B

E

68. Two tangents drawn to a circumference from the same point without this circumference are equal.

Join BE. Figure in (66.) (54.)

69. A perpendicular from a circumference

to the diameter is a mean proportional between the segments of the diameter.

A

Join AB, BC. (23.) (II. 26.)

70. If from one end of a chord a diame

B

C

ter is drawn, and from the other end a per

pendicular to this diameter, the chord is a mean proportional between the diameter and the adjacent segment of the diameter.

Join A B. (23.) (II. 25.)

.

71. The sum of the opposite sides of a circumscribed quadrilateral is equal to the sum of the other two sides. (68.)

BOOK IV.

GEOMETRY OF SPACE.

PLANES AND THEIR ANGLES.

DEFINITIONS.

1. A straight line is perpendicular to a plane when it is perpendicular to every straight line of the plane which it meets. Conversely, the plane, in this case, is perpendicular to the

line.

The foot of the perpendicular is the point in which it meets the plane.

2. A line and a plane are parallel when they cannot meet though produced indefinitely.

3. Two planes are parallel when they cannot meet though produced indefinitely.

THEOREM I.

4. A plane is determined,

1st. By a straight line and a point without that line;

2d. By three points not in the same straight line ; 3d. By two intersecting straight lines.

1st. Let the plane MN, pass- M ing through the line AB, turn upon this line as an axis until it contains the point C; the position of the plane is evidently determined; for if it is turned in

either direction it will no longer contain the point C

N

2d. If three points, A, B, C, not in the same straight line are given, any two of them, as A and B, may be joined by a straight line; then this is the same as the 1st case.

3d. If two intersecting lines A B, AC are given, any point, C, out of the line A B can be taken in the line A C; then the plane passing through the line AB and the point C contains the two lines A B and A C, and is determined by them.

5. Corollary. The intersection of two planes is a straight line; for the intersection cannot contain three points not in the same straight line, since only one plane can contain three such points.

THEOREM II.

6. Oblique lines from a point to a plane equally distant from the perpendicular are equal; and of two oblique lines unequally distant from the perpendicular, the more remote is the greater.

Let AC, AD be oblique lines drawn to the plane M N at equal distances from the perpendicular AB:

1st. ACA D'; for the triangles ABC, ABD are equal (I. 40).

D

N

2d. Let AF be more remote. From BF cut off BE = Ᏼ Ꭰ and draw A E; then A F> AE (I. 51); and AE AD AC; therefore AF> AD or AC.

7. Cor. 1. Conversely, equal oblique lines from a point to a plane are equally distant from the perpendicular; therefore they meet the plane in the circumference of a circle whose centre is the foot of the perpendicular. Of two unequal lines the

greater is more remote from the perpendicular.

8. Cor. 2. The perpendicular is the shortest distance from

a point to a plane.

E