46. CD is a chord of any circle, parallel to any given line AB; AC cuts the circle again in E, and BĚ in F: shew that DF cuts the line AB in a fixed point G. 47. In the sides AB, AC of a triangle, take two points M, N, and, in MN, a point P, such that and prove that the triangle BPC is double of the triangle AMN. 48. ABC is an inscribed triangle, AD, AE are lines drawn to BC, parallel to the tangents at C, B: shew that BE: CD :: AB2: AC2. 49. DEF is a circle inscribed in the triangle ABC, touching the sides BC, AC, AB, in D, E, F ; HIK a circle touching AB in K, and CB, CA produced in H, I: in CH take CL-CA, and in CI take CM=CB, and shew that FK=AM. 50. B, C are any two points in AD; find a point such that lines drawn from it to A, B, C, D, shall contain equal angles. 51. One side of a polygon is divided into n parts, on each of which is described a figure similar and similarly situated to the given figure: shew that the sum of the peripheries of the smaller figures is equal to that of the larger, and, if the parts be equal, that the areas of the smaller figures are together equal to th of the area of the larger. n 52. The regular inscribed hexagon is a mean proportional between the inscribed and circumscribed equilateral triangles of any circle. 53. If on the diameter of a semicircle two equal semicircles be described, and in the space between the three circumferences a circle be inscribed, its diameter will be to that of the equal circles :: 2 : 3. 54. If from the angles C, D of a quadrilateral, ABCD, and from E, the intersection of its diagonals, perpendiculars CF, DG, EH, be dropped upon AB, its 55. If Aa, Bb, Ce be drawn, bisecting the angles of a triangle, shew that 56. Given a, b, c the lengths of the chords of three arcs, which together make up a semicircle, shew that its diameter may be found from the equation x3- (a2+b2+c2) x=2abc. 57. If P, p be the areas of regular polygons of n sides, circumscribing, and inscribed in, a given circle, and Q, q the areas of corresponding polygons of 2n sides, shew that P:q:: q:p, and P: Q :: p+q: 2p. 58. Draw a perpendicular to two given lines, not in the same plane. 59. Two planes being perpendicular to each other, draw a third, perpendicular to both. 60. From two given points, draw equal lines to the same point of a given line, not in the same plane with them. 61. If three lines, not in the same plane, be equal and parallel, shew that the triangles formed by joining their adjacent extremities are equal and their planes parallel. 62. If two lines are parallel, the common section of any two planes passing through them will be parallel to either. 63. Having three points given in a plane, find a point above the plane equidistant from either of them. 64. Describe a circle which shall touch two given planes and pass through a given point. 65. If a line be perpendicular to a plane, its projection on any other plane will be perpendicular to the line of intersection of the planes. 66. Two points are taken on two walls, which meet at an angle: find the shortest line which joins them. 67. B, D are points at equal distances from the ends of the arc of a quadrant AOC, and BG, DH are perpendiculars on OC: shew that the figure BGHD is equal to the sector OBD. 68. If semicircles be described on any two segments of the diameter of a circle, the area between the three circumferences will be equal to the area of a circle, whose diameter is the mean proportional between the segments. 69. Similar arcs of different circles are as the radii, and similar sectors as the squares of the radii.* 70. Shew that the angles at the centre (or circumference) in different circles are proportional directly to the arcs subtending them and inversely to the radii; and hence that if a circle roll within another of twice its size, any point in its circumference will trace out a diameter of the first. 71. Sectors of different circles are equal, when their angles are inversely as the squares of their radii. 72. Similar segments of circles are proportional to the squares of the lines on which they stand. Hence shew that, if ABC be an isosceles triangle, right-angled at A, BDEC a semicircle on BC, and BFC a circle described with centre A, and radius AB, the segment BCF is equal to the segments BAD, ACE. 73. Any sector of a circle is equal to a triangle, whose height is the radius, and base the subtending arc. * It may be assumed that in any circle, semicircumference: radius :: area: square of radius; hence since the areas of different circles are proportional to the squares of the radii (x11. 2), their circumferences are proportional to the radii. 74. If AB be a circular arc, centre O, and AD be drawn perpendicular to BO, and the arc AC be taken equal to AD, then the sector BOC is equal to the segment ACB. 75. The difference of two similar sectors AOB, A'OB', equals the area of a rectangle, whose sides are AA', the difference of the radii, and ab, a concentric and similar arc midway between AB and A'B'. 76. If semicircles be described on the three sides of a right-angled triangle towards the interior, the difference between the sum of the circular segments thus standing upon the exterior of the sides and the sum of those upon the segments of the base, equals the space intercepted by the circumferences on the sides. 77. If a line be placed in a circle, and on the radius through one extremity a circle be described, the segments of the two circles cut off by the line will be in the ratio of 4 1. 78. Semicircles OEDA, OFDB, are described on the radii of a quadrant, OACB: shew that (i) A, B, D are in a line; (ii) the areas ACBD, OEDF are equal; (iii) the areas OFDA, OEDB, are each one-fourth of the square of the radius. 79. AB, CD are diameters of a circle, intersecting at right angles in O: with radius DA or DB describe an arc AEB, and shew that the areas of the lune AEBC and of the triangle DAB are equal. 80. AD is perpendicular on the hypothenuse of a right-angled triangle ABC: shew that the circles inscribed in the triangles ABD, ACD, will be proportional to those triangles. 81. A series of circles are described, touching each other successively and each of two given lines: if OA be the distance of the centre of the outermost circle from the intersection of the lines and OB its radius, shew that the sum of all the circles: outermost circle :: (OA+OB)2 : 40A. OB. 82. If the diameter AB of a circle be divided into n equal parts, in the points P1, P2, &c., and upon AP1, AP2, &c. semicircles be described on one side of the diameter, and also upon BP1, BP2, &c. on the other side, the perimeter of any one of the figures AP-1BP is equal to the circumference of the circle, and its area -th of the area of the circle. 1 n 83. Upon the three sides of a triangle right-angled at A, describe semicircles towards the same parts, AEB, ADC, BDC; and shew that the difference of the figures ADBE, CD, is equal to the triangle ABC. 84. The sides of a square are bisected, and by joining the bisections another is inscribed, another in this, and so on shew that all the inscribed squares are together equal to the given one. 85. Shew that all lines drawn from an external point to touch a given sphere are equal; and thence that, if a tetrahedron can have a sphere inscribed touching its six edges, the sum of every two opposite edges is the same. 86. There can be only five regular solids. 87. If Aa, Bb, Cc, be lines drawn from the angles of a triangle to the opposite sides, (i) bisecting the sides, (ii) perpendicular to the sides, (iii) bisecting the angles, (iv) to the points of contact of inscribed circle, then, in each case, Ab. Bc. Ca Ac. Ba. Cb. = 88. If Aa, Bb, Cc, be lines drawn from the angles of a triangle to the sides through any point O, then Ab. Bc. Ča Ac. Ba. Cb; and, conversely, if this equality obtain, the lines Aa, Bb, Cc, will pass through a point. 89. If two sides AB, AC of a triangle are divided proportionally in c, b, and the third, BC, bisected in a, the lines Aa, Bb, Cc, will meet in a point. 90. In 6. 2, fig. 1, if BE, CD meet in F, shew that |