THEOREM 28. If a triangle and a parallelogram stand on the same

base and between the same parallels, the area of the triangle

is half that of the parallelogram

CONSTRUCTION. To construct a triangle equivalent to a given

quadrilateral

EXERCISES

Chapter VIII.

THE THEOREM OF PYTHAGORAS.

THEOREM 29. In a right-angled triangle, the square on the hypo-

tenuse is equal to the sum of the squares on the sides con-

taining the right angle

THEOREM 30. If a triangle is such that the square on one side is

equal to the sum of the squares on the other two sides, then

the angle contained by these two sides is a right angle

NOTE ON ALGEBRA IN CONNECTION WITH GEOMETRY

.

ILLUSTRATIONS OF ALGEBRAICAL IDENTITIES BY MEANS OF

GEOMETRICAL FIGURES

(A) (a+b) k=ak+bk

(B) (a+b)2=a2+b2+2ab

(C) (a−b)2=a2+b2-2ab

(D) a2-b2=(a+b) (a - b)

EXTENSIONS OF PYTHAGORAS' THEOREM.

THEOREM 31. In an obtuse-angled triangle, the square on the side

opposite to the obtuse angle is equal to the sum of the squares

on the sides containing the obtuse angle plus twice the rect-

angle contained by one of those sides and the projection on it

of the other

THEOREM 32. In any triangle, the square on the side opposite to

an acute angle is equal to the sum of the squares on the sides

containing that acute angle minus twice the rectangle con-

tained by one of those sides and the projection on it of the

other

APOLLONIUS' THEOREM.

THEOREM 33. In any triangle, the sum of the squares on any two

sides is equal to twice the square on half the third side to-

gether with twice the square on the median which bisects the

third side.

EXERCISES

Chapter IX.

Loci.

PAGE

NOTE ON LOCI

THEOREM 34. The locus of a point which is equidistant from two

fixed points is the perpendicular bisector of the straight line

joining the two fixed points

THEOREM 35. The locus of a point which is equidistant from two

intersecting straight lines consists of the pair of straight lines

which bisect the angles between the two given lines

EXERCISES

Chapter X.

CHORDS, ARCS, TANGENTS.

THEOREM 36. A straight line, drawn from the centre of a circle to

bisect a chord which is not a diameter, is at right angles to

the chord;

Conversely, the perpendicular to a chord from the centre bisects

the chord

.

74

if

COR. 1. The perpendicular bisector of a chord of a circle will,

produced, pass through the centre of the circle

COR. 2. The locus of the centres of circles through two given

points is the perpendicular bisector of the line joining the

points

THEOREM 37. There is one circle, and one only, which passes

through three given points not in a straight line

COR. 1. Two circles cannot intersect in more than two points

COR. 2. The perpendicular bisectors of AB, BC, and CA meet in

a point.

77

77

†THEOREM 38. In equal circles (or, in the same circle)

(i) if two arcs subtend equal angles at the centres, they

are equal.

(ii) Conversely, if two arcs are equal, they subtend equal

angles at the centres

78

THEOREM 39. In equal circles (or, in the same circle)

(i) if two chords are equal, they cut off equal arcs.

(ii) Conversely, if two arcs are equal, the chords of the arcs

are equal

79

THEOREM 40. In equal circles (or, in the same circle)

(i) equal chords are equidistant from the centres.

(ii) Conversely, chords that are equidistant from the centres

are equal

80

Some examining bodies do not ask for proofs of theorems marked ‡. In each
case the various examination schedules should be consulted.

THEOREM 41. The tangent at any point of a circle and the radius

through the point are perpendicular to one another

COR. A straight line drawn through the point of contact of a

tangent at right angles to the tangent will, if produced, pass

through the centre of the circle

THEOREM 42. The two tangents to a circle from an external point

are equal.

THEOREM 43. If two circles touch, the point of contact lies in the

straight line through the centres

PAGE

81

82

82

COR. If two circles touch externally or internally, the distance

between their centres is equal respectively to the sum or

difference of their radii.

CONSTRUCTIONS. Circumscribed and inscribed circles

Circumscribed and inscribed polygons

38

83

84

84

86

86

EXERCISES

Chapter XI.

THEOREM 44. The angle which an arc of a circle subtends at the

centre is double that which it subtends at any point on the

remaining part of the circumference

THEOREM 45. Angles in the same segment of a circle are equal

THEOREM 46. If the line joining two points subtends equal angles

at two other points on the same side of it, the four points lie

on a circle

92

92

COR. The locus of the vertices of triangles with equal vertical

angles standing on the same side of a common base is an

arc of a circle

THEOREM 47. The opposite angles of any quadrilateral inscribed

in a circle are supplementary .

COR. If a side of a quadrilateral inscribed in a circle is produced,

the exterior angle so formed is equal to the interior opposite

angle of the quadrilateral

THEOREM 48. If a pair of opposite angles of a quadrilateral are

supplementary, its vertices are concyclic.

THEOREM 49. The angle in a major segment is acute; the angle

in a semicircle is a right angle; and the angle in a minor

segment is obtuse

THEOREM 50. The circle described on the hypotenuse of a right-

angled triangle as diameter passes through the opposite vertex

PAGE

THEOREM 51. If a straight line touch a circle, and from the point

of contact a chord be drawn, the angles which this chord

makes with the tangent are equal to the angles in the al-

ternate segments

CONSTRUCTIONS. Constant-angle locus

97

98

PROPORTIONAL DIVISION OF STRAIGHT LINES.

NOTE ON PROPORTIONAL DIVISION.

NOTE ON INTERNAL AND EXTERNAL DIVISION

109

110

111

THEOREM 52. If a straight line is drawn parallel to one side of a

triangle, the other two sides are divided proportionally

COR. If two straight lines are cut by a series of parallel straight

lines, the intercepts on the one have to one another the

same ratios as the corresponding intercepts on the other 111

NOTE ON INCOMMENSURABLES.

THEOREM 53. If H, K are points in the sides AB, AC of a triangle

AH AK

112

ABC, such that

AB AC'

then HK is parallel to BC

COR. 2. If a straight line divides the sides of a triangle propor-

tionally, it is parallel to the base of the triangle .

SIMILAR TRIANGLES.

THEOREM 54. If two triangles are equiangular, their corresponding

sides are proportional

114

THEOREM 55. If two triangles have their sides proportional, they

are equiangular with one another

115

THEOREM 56. If two triangles have one angle of the one equal to

one angle of the other and the sides about these equal angles

proportional, the triangles are similar

116

NOTE ON THEOREMS ON SIMILARITY OF TRIANGLES

116

THEOREM 57. The ratio of the areas of similar triangles is equal

to the ratio of the squares on corresponding sides

COR. 1. If two As ABC, XYZ have only one angle B=one angle Y,

ΔΑΒΕ BC. BA

then

PAGE

117

118

COR. 2. The ratio of the areas of similar figures is equal to the

ratio of the squares on corresponding sides.

NOTE ON THE RATIO OF AREAS OF TRIANGLES

THEOREM 58 (i). The internal bisector of an angle of a triangle

divides the opposite side internally in the ratio of the sides

containing the angle

118

118

118

119

THEOREM 58 (ii). The external bisector of an angle of a triangle

divides the opposite side externally in the ratio of the sides

containing the angle

THEOREM 59 (i). If two chords of a circle intersect inside the circle,

the rectangle contained by the segments of one is equal to the

rectangle contained by the segments of the other

COR. The rectangles contained by the segments of all chords of

a circle through the same point are equal

THEOREM 59 (ii). If two chords of a circle intersect outside the

circle, the rectangle contained by the segments of one is

equal to the rectangle contained by the segments of the other 121

COR. If a chord of a circle is produced to meet a tangent, the

120

120

square on the tangent from the point of intersection is equal

to the rectangle contained by the segments of the chord

THEOREM 60. If a perpendicular is drawn from the right angle of

a right-angled triangle to the hypotenuse, the triangles on each

side of the perpendicular are similar to the whole triangle and

to one another

CONSTRUCTIONS. To divide a line internally, or externally, in a

given ratio

To find the fourth proportional to three given straight lines.

To find the mean proportional between two given straight lines

EXERCISES

Chapter XIII.

AN INTRODUCTION TO MORE ADVANCED GEOMETRY

Revision Papers

Miscellaneous Exercises .

Index and List of Definitions