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50. Describe a circle which shall pass through a given point and which shall touch a given straight line in a given point.

51. Draw a straight line which shall touch a given circle, and make a given angle with a given straight line.

52. Describe a circle the circumference of which shall pass through a given point and touch a given circle in a given point.

53. Describe a circle with a given center, such that the circle so described and a given circle may touch one another internally.

54. Describe the circles which shall pass through a given point and touch two given straight lines.

55. Describe a circle with a given center, cutting a given circle in the extremities of a diameter.

56. Describe a circle which shall have its center in a given straight liņe, touch another given line, and pass through a fixed point in the first given line.

57. The center of a given circle is equidistant from two given straight lines; to describe another circle which shall touch the two straight lines and shall cut off from the given circle a segment containing an angle equal to a given rectilineal angle.

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58. If any two circles, the centers of which are given, intersect each other, the greatest line which can be drawn through either point of intersection and terminated by the circles, is independent of the diameters of the circles.

59. Two equal circles intersect, the lines joining the points in which any straight line through one of the points of section, which meets the circles with the other point of section, are equal.

60. Draw through one of the points in which any two circles cut one another, a straight line which shall be terminated by their circumferences and bisected in their point of section.

61. Describe two circles with given radii which shall cut each other, and have the line between the points of section equal to a given line.

62. Two circles cut each other, and from the points of intersection straight lines are drawn parallel to one another, the portions intercepted by the circumferences are equal.

63. ACB, ADB are two segments of circles on the same base AB, take any point Cin the segment ACB; join AC, BC, and produce them to meet the segment ADB in D and E respectively: shew that the arc DE is constant.

64. ADB, ACB, are the arcs of two equal circles cutting one another in the straight line AB, draw the chord ACD cutting the inner circumference in C and the outer in D, such that AD and DB together may be double of AC and CB together.

65. If from two fixed points in the circumference of a circle, straight lines be drawn intercepting a given arc and meeting without the circle, the locus of their intersections is a circle.

66. If two circles intersect, the common chord produced bisects the common tangent.

67. Shew that, if two circles cut each other, and from any point in the straight line produced, which joins their intersections, two tangents be drawn, one to each circle, they shall be equal to one another.

68. Two circles intersect in the points A and B ; through A and B any two straight lines CAD, EBF, are drawn cutting the circles in the points C, D, E, F; prove that CE is parallel to DF.

69. Two equal circles are drawn intersecting in the points A and B, a third circle is drawn with center A and any radius not greater than AB intersecting the former circles in D and C. Shew that the three points, B, C, D lie in one and the same straight line.

70. If two circles cut each other, the straight line joining their centers will bisect their common chord at right angles.

71. Two circles cut one another; if through a point of intersection a straight line is drawn bisecting the angle between the diameters at that point, this line cuts off similar segments in the two circles.

72. ACB, APB are two equal circles, the center of APB being on the circumíerence of ACB, AB being the common chord, if any chord AC of ACB be produced to cut APB in P, the triangle PBC is equilateral.

VII. 73. If two circles touch each other externally, and two parallel lines be drawn, so touching the circles in points A and B respectively that neither circle is cut, then a straight line AB will pass through the point of contact of the circles.

74. A common tangent is drawn to two circles which touch each other externally; if a circle be described on that part of it which lies between the points of contact, as diameter, this circle will pass through the point of contact of the two circles, and will touch the line which joins their centers.

75. If two circles touch each other externally or internally, and parallel diameters be drawn, the straight line joining the extremities of these diameters will pass through the point of contact.

76. If two circles touch each other internally, and any circle be described touching both, prove that the sum of the distances of its center from the centers of the two given circles will be invariable.

77. If two circles touch each other, any straight line passing through the point of contact, cuts off similar parts of their circumferences.

78. Two circles touch each other externally, the diameter of one being double of the diameter of the other; through the point of contact any line is drawn to meet the circumferences of both; shew that the part of the line which lies in the larger circle is double of that in the smaller.

79. If a circle roll within another of twice its size, any point in its circumference will trace out a diameter of the first.

80. With a given radius, to describe a circle touching two giren circles.

81. Two equal circles touch one another externally, and through the point of contact chords are drawn, one to each circle, at right angles to each ; prove that the straight line joining the other extremities of these chords is equal and parallel to the straight line joining the centres of the circles.

82. Two circles can be described, each of which shall touch a given circle, and pass through two given points outside the circle; shew that the angles which the two given points subtend at the two points of contact, are one greater and the other less than that which they subtend at any other point in the given circle.

VIII. 83. Draw a straight line which shall touch two given circles ; (1) on the same side ; (2) on the alternate sides.

84. If two circles do not touch each other, and a segment of the line joining their centers be intercepted between the convex circumferences, any circle whose diameter is not less than that segment may be so placed as to touch both the circles.

85. Given two circles : it is required to find a point from which tangents may be drawn to each, equal to two given straight lines.

86. Two circles are traced on a plane; draw a straight line cutting them in such a manner that the chords intercepted within the circles shall have given lengths.

87. Draw a straight line which shall touch one of two given circles and cut off a given segment from the other. Of how many solutions does this problem admit?

88. If from the point where a common tangent to two circles meets the line joining their centers, any line be drawn cutting the circles, it will cut off similar segments.

89. To find a point P, so that tangents drawn from it to the outsides of two equal circles which touch each other, may contain an angle equal to a given angle. : *90. Describe a circle which shall touch a given straight line at a given point, and bisect the circumference of a given circle.

91. A circle is described to pass through a given point and cut a given circle orthogonally, shew that the locus of the center is a certain straight line.

92. Through two given points to describe a circle bisecting the circumference of a given circle.

93. Describe a circle through a given point, and touching a given straight line, so that the chord joining the given point and point of contact, may cut off a segment containing a given angle.

94. To describe a circle through two given points to cut a straight line given in position, so that a diameter of the circle drawn through the point of intersection, shall make a given angle with the line.

95. Describe a circle which shall pass through two given points and cut a given circle, so that the chord of intersection may be of a given length.

IX. 96. The circumference of one circle is wholly within that of another. Find the greatest and the least straight lines that can be drawn touching the former and terminated by the latter.

97. Draw a straight line through two concentric circles, so that the chord terminated by the exterior circumference may be double that terminated by the interior. What is the least value of the radius of the interior circle for which the problem is possible ?

98. If a straight line be drawn cutting any number of concentric circles, shew that the segments so cut off are not similar.

99. If from any point in the circumference of the exterior of two concentric circles, two straight lines be drawn touching the interior and meeting the exterior; the distance between the points of contact will be half that between the points of intersection.

100. Shew that all equal straight lines in a circle will be touched by another circle.

101. Through a given point draw a straight line so that the part intercepted by the circumference of a circle, shall be equal to a given straight line not greater than the diameter.

102. Two circles are described about the same center, draw a chord to the outer circle, which shall be divided into three equal parts by the inner one. How is the possibility of the problem limited ?

103. Find a point without a given circle from which if two tangents be drawn to it, they shall contain an angle equal to a given angle, and shew that the locus of this point is a circle concentric with the given circle.

104. Draw two concentric circles such that those chords of the outer circle which touch the inner, may be equal to its diameter.

105. Find a point in a given straight line from which the tangent drawn to a given circle, is of given length.

106. If any number of chords be drawn in the inner of two concentric circles, from the same point A in its circumference, and each of the chords be then produced beyond A to the circumference of the outer circle, the rectangle contained by the whole line so produced and the part of it produced, shall be constant for all the cases.

X. 107. The circles described on the sides of any triangle as diameters will intersect in the sides, or sides produced, of the triangle.

108. The circles which are described upon the sides of a rightangled triangle as diameters, meet the hypotenuse in the same point; and the line drawn from the point of intersection to the center of either of the circles will be a tangent to the other circle.

109. If on the sides of a triangle circular arcs be described containing angles whose sum is equal to two right angles, the triangle formed by the lines joining their centers, has its angles equal to those in the segments.

110. The perpendiculars let fall from the three angles of any triangle upon the opposite sides, intersect each other in the same point.

111. If AD, CE be drawn perpendicular to the sides BC, AB of

the triangle ABC, prove that the rectangle contained by BC and BD, is equal to the rectangle contained by BA and BE.

112. The lines which bisect the vertical angles of all triangles on the same base and with the same vertical angle, all intersect in one point.

113. Of all triangles on the same base and between the same parallels, the isosceles has the greatest vertical angle.

114. It is required within an isosceles triangle to find a point such, that its distance from one of the equal angles may be double its distance from the vertical angle.

115. To find within an acute-angled triangle, a point from which, if straight lines be drawn to the three angles of the triangle, they shall make equal angles with each other.

116. A flag-staff of a given height is erected on a tower whose height is also given: at what point on the horizon will the flag-staff appear under the greatest possible angle?

117. A ladder is gradually raised against a wall ; find the locus of its middle point.

118. The triangle formed by the chord of a circle (produced or not), the tangent at its extremity, and any line perpendicular to the diameter through its other extremity, will be isosceles.

119. AD, BE are perpendiculars from the angles A and B on the opposite sides of a triangle, BF perpendicular to ED or ED produced; shew that the angle FBD=EBA.

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120. If three equal circles have a common point of intersection, prove that a straight line joining any two of the points of intersection, will be perpendicular to the straight line joining the other two points of intersection.

121. Two equal circles cut one another, and a third circle touches each of these two equal circles externally; the straight line which joins the points of section will, if produced, pass through the center of the third circle.

122. A number of circles touch each other at the same point, and a straight line is drawn from it cutting them: the straight lines joining each point of intersection with the center of the circle will be all parallel

123. If three circles intersect one another, two and two, the three chords joining the points of intersection shall all pass through one

point."

124. If three circles touch each other externally, and the three common tangents be drawn, these tangents shall intersect in a point equidistant from the points of contact of the circles.

125. If two equal circles intersect one another in A and B, and from one of the points of intersection as a center, a circle be described which shall cut both of the equal circles, then will the other point of intersection, and the two points in which the third circle cuts the other two on the same side of AB, be in the same straight line.

XII. 126. Given the base, the vertical angle, and the difference of the sides, to construct the triangle.

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