difference of the parts into which it divides the diameter that bisects it. 49. If two sides of a regular octagon, between which two others lie, be produced to meet, each of the produced parts is equal to a side of the octagon, together with the diagonal of a square, described on the side. 50. The perimeter of a triangle is to the base, as the perpendicular to the radius of the inscribed circle. 51. From a given point without a given circle, to draw a straight line cutting the circle, so that the external and internal parts may be in a given ratio. 52. Each of the complements of the parallelograms, about the diagonal of a parallelogram, is a mean proportional between those parallelograms. 53. Given the ratio of two straight lines, and the difference of their squares; to find them. 54. The square of the perimeter of a right-angled triangle is equal to twice the rectangle under the sum of the hypotenuse and one leg, and the sum of the hypotenuse and the other. 55. The quadrilateral formed by straight lines bisecting each pair of adjacent sides of a quadrilateral, is a parallelogram which is half of the quadrilateral; and straight lines, joining the points in which the sides of that parallelogram are cut by the diagonals of the primitive figure, form a quadrilateral similar to that figure and equal to a fourth of it. 56. From three given points as centres, and not in the same straight line, to describe three circles each touching the other two. Show that this admits of four solutions. 57. To add a parallelogram to a rhombus, such that the whole figure may be a parallelogram similar to the one added. 58. A straight line being given in position, and a circle in magnitude and position, it is required to describe two equal circles touching one another, and each touching the straight line and the circle. 59. The squares of the diagonals of a quadrilateral are together double of the squares of the straight lines joining the points of bisection of the opposite sides. 60. In a given rhombus to inscribe a rectangle having its sides in a given ratio. 61. If, through the vertex and the extremities of the base of a triangle, two circles be described intersecting one another in the base or its continuation, their diameters are proportional to the sides of the triangle. 62. To draw a straight line cutting two given concentric circles, so that the parts of it within them may be in a given ratio. 63. From a given point, within a given circle, or without it, to draw two straight lines to the circumference, perpendicular to one another, and in a given ratio. When will this be impossible? 64. If a straight line be divided in extreme and mean ratio, the squares of the whole and the less part are together equal to three times the square of the greater. 65. If a straight line be cut in extreme and mean ratio, and be also bisected, the square of the intermediate part, and three times the square of half the line are equal to twice the square of the greater part. 66. If the hypotenuse of a right-angled triangle be given, the side of the greatest inscribed square, standing on the hypotenuse, is one third of the hypotenuse. 67. To divide a given semicircle into two parts by a perpendicalar to the diameter, so that the radii of the circles inscribed in them, may be in a given ratio. 68. To draw a straight line parallel to the base of a triangle, making a segment of one side equal to the remote segment of the other. 69. In the figure for the 11th proposition of the second book of Euclid, the square of the diameter of the circle, passing through the points F, H, D, is six times the square of the straight line joining FD. 70. Given the base, the area, and the sum of the squares of the sides of a triangle; to construct it. 71. If, from the extremities of the hypotenuse of a right-angled triangle as centres, arcs be described passing through the right angle, the hypotenuse is divided into three segments, such that the square of the middle one is equal to twice the rectangle of the others.* 72. On a given hypotenuse to describe a right-angled triangle, such that the difference between one leg and the adjacent segment of the hypotenuse made by a perpendicular from the right angle, may be a maximum. 73. On a given hypotenuse to construct a right-angled triangle, such that one segment of the hypotenuse made by the perpendicular, from the right angle, may be equal to the sum of the perpendicular, and the other segment. 74. The square of CD (see the figure for proposition XXV, page 248) is equal to the rectangle MK.NL; and if the circles touch one another externally, CD is a mean proportional between their diameters. Also the square of FG is equal to the rectangle ML.NK. 75. On a given straight line to describe an isosceles triangle, having the vertical angle treble of each of the angles at the base. 76. If BD, CE, in the figure in page 245, be joined, the interior figure contained by the segments of the diagonals, is a regular pentagon; and if each pair of remote sides AB, DC, &c. be produced to meet, their points of intersection will be the angular points of another regular pentagon. 77. To find a point from which, if straight lines be drawn to three given points, they will be proportional to three given straight lines. 78. Given the segments into which the base of a triangle is divided by two straight lines trisecting the vertical angle; to construct it. * Compare this with Appendix, Book I. Prop. XXVII. Schol. 79. If one diagonal of a quadrilateral inscribed in a circle, be bisected by the other, the square of the latter is equal to half the sum of the squares of the four sides. 80. To divide a straight line into two parts, such that the squares of the whole and one of the parts may be double of the square of the other part. 81. Given the segments into which the base of a triangle is divided by the straight line bisecting the vertical angle, to construct the triangle so that its angle adjacent to the greater segment may be either of a given magnitude, or a maximum, and in each case to compute the remaining sides and angles. 82. To draw a straight line bisecting a given parallelogram, so that if it be produced to meet the sides produced, the external triangles will have a given ratio to the parallelogram. 83. Through a given point to draw a straight line, which, if continued, would pass through the point of intersection of two given inclined straight lines, without producing those lines to meet. 84. If, from any point in the circumference of the circle described about an equilateral triangle, chords be drawn to its three angles, the sum of their squares is equal to six times the square of the radius of the same circle, or to twice the square of a side of the triangle. 85. Given the difference of the angles at the base of a triangle, the difference of the segments into which the base is divided by the perpendicular, and the ratio of the sides; to construct the triangle. 86. Given the base and vertical angle of a triangle, to construct it, so that the line bisecting the vertical angle may be a mean proportional between the segments into which it divides the base. 87. Given two sides of a triangle, and the ratio of the base and the line bisecting the vertical angle; to construct the triangle. 88. If the vertical angle of a triangle be double of one of the angles at the base, the rectangle under the sides is equal to the rectangle under the base, and the line bisecting the vertical angle. 89. If a straight line be divided into parts, which, taken in succession, are continual proportionals, and if circles be described on the several parts as diameters, a straight line which touches two of the circles on the same side of the straight line joining their centres, will touch all the others. 90. Given the segments into which the base is divided by the straight line bisecting the vertical angle, and the angles which that straight line makes with the base; to construct the triangle. 91. To divide a given circle into two segments, such that the squares inscribed in them may be in a given ratio. 92. Through a given point in the base of a given isosceles triangle, or its continuation, to draw a straight line such that the lines intercepted on the equal sides, or their continuations between that line and the extremities of the base, may have one of the equal sides as a mean proportional between them. 93. Through a point in the circumference of a given circle, to draw two chords, such that their rectangle may be equal to a given space, and the chord joining their other extremities equal to a given straight line. 94. In any triangle the radius of the circumscribed circle is to the radius of the circle which is the locus of the vertex, when the base and the ratio of the sides are given, as the difference of the squares of those sides is to four times the area. 95. The difference of the sides of a triangle is a mean proportional between the difference of the segments into which the base is divided by the perpendicular, and the difference of those into which it is divided by the line bisecting the vertical angle. 96. Let the angles of a parallelogram which has unequal sides, be bisected by straight lines cutting the diagonals, and let the points of intersection be joined: the figure thus formed is a parallelogram, which has to the proposed parallelogram the duplicate ratio of that which the difference of the unequal sides of the latter has to their sum. 97. A circle and a point being given, it is required to describe a triangle similar to a given one, having its vertex at the given point, and its base a chord of the given circle. 98. Given the three points in which the sides of a triangle are cut by the perpendiculars from the opposite angles; to construct the triangle. 99. Given the angles of a triangle, and the lengths of three straight lines drawn from the angular points to meet in another point; to construct the triangle. 100. Given the base of a triangle, and the ratio of its sides; to construct it, so that the distance of its vertex from a given point may be a maximum or minimum. 101. To divide a circle into two segments, such that the sum of the squares inscribed in them may be equal to a given space. 102. Through a given point, with a given radius, to describe a circle bisecting the circumference of a given circle. 103. With a given radius to describe a circle bisecting the circumferences of two given circles. 104. In a right-angled triangle, the rectangle under the radius of the inscribed circle, and the radius of the circle touching the hypotenuse and the legs produced, is equal to the area. So, likewise, is the rectangle under the circles touching the legs externally, and the continuations of the other sides. 105. If three straight lines be continual proportionals, the sum of the extremes, their difference, and double the mean, will be the hypotenuse and legs of a right-angled triangle. 106. From two given centres, to describe circles having their radii in a given ratio, and the part of their common tangent, between the points of contact, equal to a given straight line. 107. In a given circle to inscribe a quadrilateral, having two of its sides equal to two given lines, and the other two in a given ratio. 108. With a given radius to describe a circle touching two given circles. When is this impossible, and when are there two, or more solutions? 109. Through a given point to describe a circle touching a given circle, and having its centre in a given straight line. When is this impossible? 110. With a given radius, to describe a circle passing through a given point, and touching a given straight line. 111. To trisect a given triangle by straight lines drawn from a given point within the triangle, one of those lines being given. 112. Given the angles and the vertex of a triangle, to describe it, so that the extremities of the base may be on two straight lines given in position. 113. To divide a given chord of a given circle into two parts, such that their rectangle may have a given ratio to the square of the straight line drawn from the point of section to a given point in the circumference. 114. To describe two circles touching each other, and each of them touching the base and one of the remaining sides of a given triangle, the ratio of their radii being given. 115. Two straight lines being given in position, and two points being given in one of them; it is required to find two points in the other, such that if two straight lines be drawn from the given points to the latter, and two from the latter to meet in the other line, the sum of all the four will be a minimum. 116. Let BEF (see the figure for proposition G of the sixth book) be a given triangle, and (VI. G, cor.) through B let the circle ABC be described, such that straight lines drawn from any point in its circumference may be proportional to BE, BF: then, if through E and F two other circles be described, the first, such that straight lines drawn from B and F, to meet in its circumference, may be proportional to EB, EF, and the second, such that straight lines drawn from E and B, to meet in its circumference, may be proportional to FE, FB; the centres of the three circles will lie in the same straight line, and there are two points through which all their circumferences pass. 117. In the figure for proposition H of the sixth book, BF : FC: AC.BD: AB.DC. 118. To describe a triangle having its sides equal to three given straight lines, and passing through three given points. 119. On a given straight line as hypotenuse, to describe a right angled triangle, such that, of the triangles into which it is divided by the perpendicular from the right angle, one may be a maximum. 120. In a given triangle ABC, (see the first figure for the second proposition of the sixth book) draw DE parallel to BC, so that BE being joined, the triangle BDE may be a maximum. 121. To describe a circle having its centre in a given line, and bisecting the circumferences of two given circles. 122. A straight line and two points equally distant from it, on the same side, being given in position, it is required to draw through the points two straight lines forming with the given line the least isosceles triangle possible, on the side on which the points are. 123. To describe a circle touching a diameter of a given circle |