63. From a given point within a given circle to draw a straight

line which shall make with the circumference an angle less than the

angle made by any other line drawn from that point.

64. To determine a point in the arc of a quadrant, from which

if lines be drawn to the centre and the point of bisection of the radius,

they shall contain the greatest possible angle.

65. If the radius of a circle be a mean proportional to two dis-

tances from the centre in the same straight line; the lines drawn

from their extremities to any point in the circumference will have

the same ratio that the distances of these points from the circum-
ference have.

66. Two circles being given in position and magnitude; to

draw a straight line cutting them so that the chords in each circle

may be equal to a given line, not greater than the diameter of the

smaller circle.

67. To determine a point in the arc of a quadrant, through

which if a tangent be drawn meeting the sides of the quadrant pro-

duced, the intercepted parts may have a given ratio.

68. If a tangent be drawn to a circle at the extremity of a chord

which cuts the diameter at right angles, and from any point in it a

perpendicular be let fall; the segment of the diameter intercepted

between that perpendicular and chord is to the intercepted part of

the tangent as the chord is to the diameter.

69. If a straight line be placed in a circle, and from its ex-

tremities perpendiculars be let fall upon any diameter; these per-

pendiculars together will have to the part of the diameter intercepted

between them, the same ratio that a line placed in the circle per-

pendicular to the former line, has to the former line itself.

70. In a circle to place a straight line of given length, so that

perpendiculars drawn to it from two given points in the circum-

ference may have a given ratio.

71. If from any point in the arc of a segment of a circle a line

be drawn perpendicular to the base; and from the greater segment

of the base and arc, parts be cut off respectively equal to the

less; the remaining part of the base shall be equal to the chord of

the remaining arc.

72. If from the point of bisection of any arc of a circle a per-

pendicular be drawn to the diameter which passes through one

extremity; it will bisect the segment of the chord cut off by the line

joining the point of bisection of the arc and the other extremity of

the diameter.

73. In a given circle to draw a chord parallel to a straight line

given in position; so that the chord and perpendicular drawn to it

from the centre may together be equal to a given line.

74. Through a given point within a given circle, to draw a

straight line such that the parts of it intercepted between that point

and the circumference may have a given ratio.

75. From two given points in the circumference of a given circle,

to draw two lines to a point in the circumference, which shall cut a

line given in position, so that the part of it intercepted by them may

be equal to a given line.

76. If a chord and diameter of a circle intersect each other at

any angle, and a perpendicular to the chord be drawn from either

extremity of it, meeting the circumference and diameter produced;

the whole perpendicular has to the part of it without the circle, the

same ratio that the greater segment of the chord has to the less.

77. If from the extremities of any chord of a circle, perpen-

diculars to it be drawn and produced to cut a diameter; and from

the points of intersection with the diameter lines be drawn to a

point in the chord so as to make equal angles with it; these lines

together will be equal to the diameter of the circle.

78. If from a point without a circle two straight lines be drawn,

one of which touches and the other cuts the circle; a line drawn

from the same point in any direction, equal to the tangent, will

be parallel to the chord of the arc intercepted by two lines drawn

from its other extremity to the former intersections of the circle.

79. If from a point without a circle two straight lines be drawn

touching it, and from one point of contact a perpendicular be drawn

to that diameter which passes through the other; this perpendicular

will be bisected by the line joining the point without the circle and

the other extremity of the diameter.

80. If any chord in a circle be bisected by another, and produced

to meet the tangents drawn from the extremities of the bisecting line;

the parts intercepted between the tangents and the circumferences

are equal.

81.. If one chord in a circle bisect another, and tangents drawn

from the extremities of each be produced to meet; the line joining

their points of intersection will be parallel to the bisected chord.

82. If from a point without a circle two lines be drawn touching

the circle, and from the extremities of any diameter lines be drawn

to the points of contact, cutting each other within the circle; the

line produced, which joins their intersection and the point without

the circle, will be perpendicular to the diameter.

83. If on opposite sides of the same extremity of the diameter of

a circle equal arcs be taken, and from the extremities of these arcs

lines be drawn to any point in the circumference, one of which cuts

the diameter, and the other the diameter produced; the distances

of the points of intersection from the extremities of the diameter

are proportional to each other.

84. If from the extremities of any chord in a circle, perpen-

diculars be drawn to a diameter, and from either extremity of that

diameter a perpendicular be drawn to the chord; it will divide it

into segments, which are respectively mean proportionals between

the segments of the diameter cut off by the perpendiculars.

85. If from any point in the diameter of a semicircle, a per-

pendicular be drawn, meeting the circumference, and on it as a

diameter a circle be described, to the centre of which a line is

drawn from the farther extremity of the diameter of the semicircle,

cutting its circumference; and through the point of intersection

another line be drawn from the extremity of the perpendicular,

meeting the diameter of the semicircle; this diameter will be divided

into three segments which are in continued proportion.

86. If from a point without a given circle, any two lines be

drawn cutting the circle; to determine a point in the circumference,

such that the sum of the perpendiculars from it upon these lines may

be equal to a given line.

87. If two circles cut each other, and any two points be taken

in the circumference of one of them, through which lines are drawn

from the points of intersection and produced to the circumference of

the other; the straight lines joining the extremities of those which

are drawn through the same point, are equal.

88. If two circles cut each other; the greatest line that can be

drawn through the point of intersection is that which is parallel to

the line joining their centres.

89. Having given the radii of two circles which cut each other,

and the distance of their centres; to draw a straight line of given

length through their point of intersection, so as to terminate in their

circumferences.

90. If two circles cut each other; to draw from one of the points

of intersection a straight line meeting the circles, so that the part of

it intercepted between the circumferences may be equal to a given

line.

91. If two circles cut each other; to draw from the point of

intersection two lines, the parts of which intercepted between the

circumferences may have a given ratio.

92. If a semicircle be described on the common chord of two

intersecting circles, and a line drawn from one extremity of the

chord, cutting the two circles; the part intercepted between the two

shall be divided by the semicircle into segments proportional to

perpendiculars drawn in those circles from the other extremity of

the chord.

93. Two circles being given, the circumference of one of which

passes through the centre of the other; to draw a chord from that

centre, such that a perpendicular let fall upon it from a given point,

may bisect that part of it which is intercepted between the cir-

cumferences.

94. If any number of circles cut each other in the same points,

and from one of these points any number of lines be drawn; the parts

of them which are intercepted between the several circumferences

have the same ratio.

95. In a given circle to place a straight line cutting two radii

which are perpendicular to each other, in such a manner that the

line itself may be trisected.

96. If a straight line be divided into any two parts, and upon

the whole line and one of the parts, as diameters, semicircles be

described; to determine a point in the less diameter, from which if

a perpendicular be drawn cutting the circumferences, and the points

of intersection and the extremities of the respective diameters be

joined, and these lines produced to meet; the parts of them without

the semicircles may have a given ratio.

97. If a straight line be divided into any two parts, and from the

point of section a perpendicular be erected, which is a mean pro-

portional between one of the parts and the whole line, and a circle

described through the extremities of the line and the perpendicular;

the whole line, the perpendicular, the aforesaid part, and a per-

pendicular drawn from its extremity to the circumference will be in

continued proportion.

98. If the tangents drawn to every two of three unequal circles

be produced till they meet, the points of intersection will be in a

straight line.

any

number

99. If from the extremities of the diameter of a circle

of chords be drawn, two and two intersecting each other in a per-

pendicular to that diameter; the lines joining the extremities of

every corresponding two will meet the diameter produced in the

same point.

100. If from a given point in the diameter of a semicircle pro-

duced, three straight lines be drawn, one of which is inclined at a

given angle to the diameter, another touches the semicircle, and the

third cuts it, in such a manner, that the distance of the given point

from the nearer extremity of the diameter, and the perpendiculars

drawn from that extremity on the three aforesaid lines may be

proportional; then will the lines, which join the extremities of the

diameter and of that part of the cutting line which is within the

circle, intersect each other in an angle equal to the given angle.