49. If the base of a triangle be bisected by the diameter of the circumscribing circle, and from the extremity of that diameter a perpendicular be let fall on the longer side, it will divide that side into segments, one of which is equal to the semisum and the other to the semidifference of the sides. 50. Given the radius of a circle, that touches two given lines not parallel, determine its centre. 51. Find a point in the diameter produced of a given circle, such that, if tangents be drawn from it to the circle, the concave part of the circumference may be double of the convex. 52. The line drawn through the bisection of any arc of a circle, parallel to its chord, is a tangent to the circle at that point: and the radius which bisects the chord of an arc bisects also the arc. 53. If, from each extremity of two adjacent arcs of a circle, lines be drawn through two given points in the opposite circumference and produced till they meet, the angles formed by these lines will be equal. 54. If from the middle point of any arc of a circle, a perpendicular be drawn to the diameter through one extremity, it will bisect the segment of the chord cut off by the line, which joins the middle point aforesaid and the other extremity of the diameter. 55. Two equal circles pass through each others' centres A, B: if any common chord CEFD, be drawn parallel to AB, shew that the figures ACEB, AFDB, are parallelograms; and, if AF be produced to meet the circumferences again in G and H, shew that EF and FH, as also CD and GH, are equal. 56. ABC is a triangle inscribed in a circle, DEF a diameter cutting BC at right angles in E: shew that the difference of the angles at B and C is double of the angle AFD. 57. ACB, ADB are arcs of equal circles on the same line AB and on the same side of it: draw any chord ACD cutting them both, and shew that BC and BD are equal. 58. If two chords in a circle intersect each other, the angle between them is half the angle at the centre subtended by an arc, which is equal to the sum or difference of the arcs intercepted by them, according as they intersect within or without the circle. Shew also that if they intersect at right angles, the sum of any two opposite arcs equals half the circumference. 59. If circles be described on the two sides of a rightangled triangle as diameters, they will be touched by a circle, whose centre is the bisection of the hypothenuse, and diameter equal to the sum of the sides. 60. If perpendiculars Aa, Bb, Cc be drawn from the angular points of a triangle ABC upon the sides, shew that they will bisect the angles of the triangle abc. 61. If a circle be described on the radius of another circle, any line drawn from the point where they meet to the outer circumference is bisected by the inner. 62. The circles described on the sides of any triangle as diameters will intersect in the sides, or sides produced, of the triangle. 63. If from the extremity of the diameter of a circle any chord be drawn, shew that its middle point will always lie in the circumference of a given circle. 64. If an equilateral triangle be inscribed in a circle, and the adjacent arcs cut off by two of its sides be bisected, the line joining the points of section will be trisected by the sides. 65. The vertical angle of any oblique-angled triangle inscribed in a circle is greater or less than a right-angle by the angle contained by the base and the diameter drawn from the extremity of the base: and no parallelogram can be inscribed in a circle except a rectangle. 66. From one extremity of a line, which cannot be produced, draw a line perpendicular to it. 67. ABC is an equilateral triangle, D, E, F the bisections of its sides; shew that DF touches the circle CDE. 68. If upon any radius of a circle a circle be described, and any other two radii be drawn cutting it, the chord of the arc intercepted by them is equal to the perpendicular from the extremity of one upon another. 69. If two circles touch each other and any two lines be drawn passing through the point of contact, the chords of the intercepted arcs will be parallel. 70. If a semicircle be described on the side of a quadrant, and from any point in the quadrantal arc, a radius be drawn, the part of it between the quadrant and the semicircle is equal to the perpendicular from the same point on the common tangent. 71. Given one angle, the side opposite, and the sum of the other two sides, construct the triangle. 72. Given the area and hypothenuse of a right-angled triangle: construct it. 73. Given the base, vertical angle, and altitude: construct the triangle. 74. Of all triangles on the same base and having the same vertical angle, the isosceles is greatest; and, conversely, of all triangles on the same base and between the same parallels, the isosceles has the greatest vertical angle. 75. Through three given points draw three lines, so as to make an equilateral triangle. 76. If two circles cut each other, draw through the point of section a line cutting both the circles and equal to a given line; and hence through three given points draw lines, so as to make a triangle equal in all respects to a given triangle. 77. Find a point within a triangle at which the three sides shall subtend equal angles: and shew that if on the three sides of a triangle towards the same parts similar segments of circles be described, and from the extremities of the base tangents to the segment upon it be drawn to cut the other circumferences, the points of intersection and the vertex will be in the same line parallel to the base. 78. AB is the diameter of a circle, CD a chord perpendicular to AB; if through any point P in CD a chord APQ be drawn, the rectangle AP, AQ is con stant. 79. Given the area, one angle, and a line drawn from one of the others bisecting the opposite side: construct the triangle. 80. Describe a circle which shall pass through two given points and touch a given line: and shew that, of all triangles upon a given base and between the same parallels, the isosceles has the greatest vertical angle. 81. Describe a circle which shall pass through two given points and touch a given line in a given point. 82. Through a given point describe a circle touching two given lines. 83. Describe a circle touching two given lines and a given circle. 84. Describe an isosceles triangle, having given the base angle and the perpendicular from it upon the opposite side. 85. If from the centre of a circle a line be drawn to any point in the chord of an arc, the square of that line together with the rectangle of the segments of the chord will be equal to the square of the radius. 86. Let the diameter of a circle cut at right angles in A a given chord, and let any other chord, BC, cut the same chord in D: then the sum of the square of AD and the rectangle of BD, CD is constant. 87. If through any point within or without a circle two lines cut each other at right angles, the sums of the squares of the two lines joining their extremities, and also the sum of the squares of the four segments, are each equal to the square of the diameter. 88. If there be any three circles in a plane, and, through the centres of each two of them, a circle be described touching the third, the lines, joining the centre of each of the circles with the point in which the circles passing through it intersect, will meet in the same point. 89. Two points are taken in the diameter of a circle at any equal distances from the centre: through one of these draw any chord, and join its extremities with the other point; and shew that the triangle thus formed has the sum of the squares of its sides invariable. 90. ABC is a triangle whose acute vertex is A; shew that the square of BC is less than the squares of AB, AC by twice the square of the line drawn from A to touch the circle on BC as diameter. 91. If any number of circles be drawn through two given points cutting a given circle, the lines which join the points of intersection shall all meet the line joining the two given points in the same point. 92. If three circles cut each other two and two, their three chords of intersection meet in a point. 93. Let ACDB be a semicircle, whose diameter is AB, and AD, BC any two chords intersecting in P; shew that AB=AD.AP+BC.BP. 94. Let a common tangent be drawn to any number of circles which touch each other internally, and with any point of this tangent as centre describe a circle cutting the other circles; then if from this centre lines be drawn through the intersections of the circles, the segments of these lines within each circle will be equal. 95. A rod of given length is moved betwixt two fixed straight lines CP, CQ; the perpendiculars from P, Q upon CP, CQ meet in R, and those from P, Q upon CQ, CP meet in S; shew that the loci of R, S are circles with common centre C. |