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. 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. XI. 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. 127. Describe a triangle, having given the vertical angle, and the segments of the base made by a line bisecting the vertical angle. 128. Given the perpendicular height, the vertical angle and the sum of the sides, to construct the triangle. 129. Construct a triangle in which the vertical angle and the difference of the two angles at the base shall be respectively equal to two given angles, and whose base shall be equal to a given straight line. 130. Given the vertical angle, the difference of the two sides containing it, and the difference of the segments of the base made by a perpendicular from the vertex; construct the triangle. 131. Given the vertical angle, and the lengths of two lines drawn from the extremities of the base to the points of bisection of the sides, to construct the triangle. 132. Given the base, and vertical angle, to find the triangle whose area is a maximum. 133. Given the base, the altitude, and the sum of the two remaining sides; construct the triangle. 134. Describe a triangle of given base, area, and vertical angle. 135. Given the base and vertical angle of a triangle, find the locus of the intersection of perpendiculars to the sides from the extremities of the base. XIII. 136. Shew that the perpendiculars to the sides of a quadrilatera. inscribed in a circle from their middle points intersect in a fixed point. 137. The lines bisecting any angle of a quadrilateral figure inscribed in a circle, and the opposite exterior angle, meet in the circumference of the circle. 138. If two opposite sides of a quadrilateral figure inscribed in a circle be equal, prove that the other two are parallel. 139. The angles subtended at the center of a circle by any two opposite sides of a quadrilateral figure circumscribed about it, are together equal to two right angles. 140. Four circles are described so that each may touch internally three of the sides of a quadrilateral figure, or one side and the adjacent sides produced; shew that the centers of these four circles will all lie in the circumference of a circle. 141. One side of a trapezium capable of being inscribed in a given circle is given, the sum of the remaining three sides is given; and also one of the angles opposite to the given side: construct it. 142. If the sides of a quadrilateral figure inscribed in a circle be produced to meet, and from each of the points of intersection a straight line be drawn, touching the circle, the squares on these tangents are together equal to the square on the straight line joining the points of intersection. 143. If a quadrilateral figure be described about a circle, the sums of the opposite sides are equal; and each sum equal to half the perimeter of the figure. 144. A quadrilateral ABCD is inscribed in a circle, BC and DC are produced to meet AD and AB produced in E and F. The angles ABC and ADC are together equal to AFC, AEB, and twice the angle BAC. 145. If the hypotenuse AB of a right-angled triangle ABC be bisected in D, and EDF drawn perpendicular to AB, and DE, DF cut off each equal to DA, and CE, CF joined, prove that the last two lines will bisect the angle at C and its supplement respectively. 146. ABCD is a quadrilateral figure inscribed in a circle. Through its angular points tangents are drawn so as to form another quadrilateral figure FBLCHDEA circumscribed about the circle. Find the relation which exists between the angles of the exterior and the angles of the interior figure. 147. The angle contained by the tangents drawn at the extremities of any chord in a circle is equal to the difference of the angles in segments made by the chord: and also equal to twice the angle contained by the same chord and a diameter drawn from either of its extremities. 148. If ABCD be a quadrilateral figure, and the lines AB, AC, AD be equal, shew that the angle BAĎ is double of CBD and CDB together. 149. Shew that the four lines which bisect the interior angles of a quadrilateral figure, form by their intersections, a quadrilateral figure which can be inscribed in a circle. 150. In a quadrilateral figure ABCD is inscribed a second quadrilateral by joining the middle points of its adjacent sides; a third is similarly inscribed in the second, and so on. Shew that each of the series of quadrilaterals will be capable of being inscribed in a circle if the first three are so. Shew also that two at least of the opposite sides of ABCD must be equal, and that the two squares upon these sides are together equal to the sum of the squares upon the other two. XIV. 151. If from any point in the diameter of a semicircle, there be drawn two straight lines to the circumference, one to the bisection of the circumference, the other at right angles to the diameter, the squares upon these two lines are together double of the square upon the semi-diameter. 152. If from any point in the diameter of a circle, straight lines be drawn to the extremities of a parallel chord, the squares on these lines are together equal to the squares on the segments into which the diameter is divided. 153. From a given point without a circle, at a distance from the circumference of the circle not greater than its diameter, draw a straight line to the concave circumference which shall be bisected by the convex circumference. 154. If any two chords be drawn in a circle perpendicular to each other, the sum of their squares is equal to twice the square on the diameter diminished by four times the square on the line joining the center with their point of intersection. |