51. If from one angle 4 of a parallelogram a straight line be drawn cutting the diagonal in E and the sides in P, Q, shew that

AEPE. EQ.

52. The diagonals of a trapezium, two of whose sides are parallel, cut one another in the same ratio.

VII.

53. In a given circle place a straight line parallel to a given straight line, and having a given ratio to it; the ratio not being greater than that of the diameter to the given line in the circle.

54. In a given circle place a straight line, cutting two radii which are perpendicular to each other, in such a manner, that the line itself may be trisected.

55. AB is a diameter, and P any point in the circumference of a circle; AP and BP are joined and produced if necessary; if from any point C of AB, a perpendicular be drawn to AB meeting AP and BP in points D and E respectively, and the circumference of the circle in a point F, shew that CD is a third proportional of CE and CF.

56. If from the extremity of a diameter of a circle tangents be drawn, any other tangent to the circle terminated by them is so divided at its point of contact, that the radius of the circle is a mean proportional between its segments.

57. From a given point without a circle, it is required to draw a straight line to the concave circumference, which shall be divided in a given ratio at the point where it intersects the convex circumference.

58. From what point in a circle must a tangent be drawn, so that a perpendicular on it from a given point in the circumference may be cut by the circle in a given ratio?

59. Through a given point within a given circle, to draw a straight line such that the parts of it intercepted between that point and the circumference, may have a given ratio.

60. Let the two diameters AB, CD, of the circle ADBC be at right angles to each other, draw any chord EF, join CE, CF, meeting AB in G and H; prove that the triangles CGH and CEF are similar.

61. A circle, a straight line, and a point being given in position, required a point in the line, such that a line drawn from it to the given point may be equal to a line drawn from it touching the circle. What must be the relation among the data, that the problem may become porismatic, i.e. admit of innumerable solutions ?

VIII.

62. Prove that there may be two, but not more than two, similar triangles in the same segment of a circle.

63. If as in Euclid vi. 3, the vertical angle BAC of the triangle BAC be bisected by AD, and BA be produced to meet CE drawn parallel to AD in E; shew that AD will be a tangent to the circle described about the triangle EAC.

64. If a triangle be inscribed in a circle, and from its vertex, lines be drawn parallel to the tangents at the extremities of its base, they will cut off similar triangles.

65. If from any point in the circumference of a circle perpendiculars be drawn to the sides, or sides produced, of an inscribed triangle; shew that the three points of intersection will be in the same straight line.

66. If through the middle point of any chord of a circle, two chords be drawn, the lines joining their extremities shall intersect the first chord at equal distances from its extremities.

67. If a straight line be divided into any two parts, to find the locus of the point in which these parts subtend equal angles.

68. If the line bisecting the vertical angle of a triangle be divided into parts which are to one another as the base to the sum of the sides, the point of division is the center of the inscribed circle.

69. The rectangle contained by the sides of any triangle is to the rectangle by the radii of the inscribed and circumscribed circles, as twice the perimeter is to the base.

70. Shew that the locus of the vertices of all the triangles constructed upon a given base, and having their sides in a given ratio, is a circle. 71. If from the extremities of the base of a triangle, perpendiculars be let fall on the opposite sides, and likewise straight lines drawn to bisect the same, the intersection of the perpendiculars, that of the bisecting lines, and the center of the circumscribing circle, will be in the same straight line.

IX.

72. If a tangent to two circles be drawn cutting the straight line which joins their centers, the chords are parallel which join the points of contact, and the points where the line through the centers cuts the circumferences.

73. 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.

74. If two circles touch each other externally and also touch a straight line, the part of the line between the points of contact is a mean proportional between the diameters of the circles.

75. If from the centers of each of two circles exterior to one another, tangents be drawn to the other circles, so as to cut one another, the rectangles of the segments are equal.

76. If a circle be inscribed in a right-angled triangle and another be described touching the side opposite to the right angle and the produced parts of the other sides, shew that the rectangle under the radii is equal to the triangle, and the sum of the radii equal to the sum of the sides which contain the right angle.

77. If a perpendicular be drawn from the right angle to the hypotenuse of a right-angled triangle, and circles be inscribed within the two smaller triangles into which the given triangle is divided, their diameters will be to each other as the sides containing the right angle.

X.

78. Describe a circle passing through two given points and touching a given circle.

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

80. Through a given point draw a circle touching two given circles.

81. Describe a circle to touch two given right lines and such that a tangent drawn to it from a given point, may be equal to a given line. 82. Describe a circle which shall have its center in a given line, and shall touch a circle and a straight line given in position.

XI.

83. Given the perimeter of a right-angled triangle, it is required to construct it, (1) If the sides are in arithmetical progression. (2) If the sides are in geometrical progression.

84. Given the vertical angle, the perpendicular drawn from it to the base, and the ratio of the segments of the base made by it, to construct the triangle.

85. Apply (vI. c.) to construct a triangle; having given the vertical angle, the radius of the inscribed circle, and the rectangle contained by the straight lines drawn from the center of the circle to the angles at the base.

86. Describe a triangle with a given vertical angle, so that the line which bisects the base shall be equal to a given line, and the angle which the bisecting line makes with the base shall be equal to a given angle.

87. Given the base, the ratio of the sides containing the vertical angle, and the distance of the vertex from a given point in the base; to construct the triangle.

88. Given the vertical angle and the base of a triangle, and also a line drawn from either of the angles, cutting the opposite side in å given ratio, to construct the triangle.

89. Upon the given base AB construct a triangle having its sides in a given ratio and its vertex situated in the given indefinite line CD. 90. Describe an equilateral triangle equal to a given triangle. 91. Given the hypotenuse of a right-angled triangle, and the side of an inscribed square. Required the two sides of the triangle.

92. To make a triangle, which shall be equal to a given triangle, and have two of its sides equal to two given straight lines; and shew that if the rectangle contained by the two straight lines be less than twice the given triangle, the problem is impossible.

XII.

93. Given the sides of a quadrilateral figure inscribed in a circle, to find the ratio of its diagonals.

94. The diagonals AC, BD, of a trapezium inscribed in a circle, cut each other at right angles in the point E;

the rectangle AB.BC: the rectangle AD.DC :: BE: ED.

XIII.

95. In any triangle, inscribe a triangle similar to a given triangle. 96. Of the two squares which can be inscribed in a right-angled triangle, which is the greater?

97. From the vertex of an isosceles triangle two straight lines

drawn to the opposite angles of the square described on the base, cut the diagonals of the square in E and F: prove that the line EF is parallel to the base.

98. Inscribe a square in a segment of a circle.

99. Inscribe a square in a sector of a circle, so that the angular points shall be one on each radius, and the other two in the circumference.

100. Inscribe a square in a given equilateral and equiangular pentagon.

101. Inscribe a parallelogram in a given triangle similar to a given parallelogram.

102. If any rectangle be inscribed in a given triangle, required the locus of the point of intersection of its diagonals.

103. Inscribe the greatest parallelogram in a given semicircle. 104. In a given rectangle inscribe another, whose sides shall bear to each other a given ratio.

105. In a given segment of a circle to inscribe a similar segment. 106. The square inscribed in a circle is to the square inscribed in the semicircle :: 5 : 2.

107. If a square be inscribed in a right-angled triangle of which one side coincides with the hypotenuse of the triangle, the extremities of that side divide the base into three segments that are continued proportionals.

108. The square inscribed in a semicircle is to the square inscribed in a quadrant of the same circle :: 8: 5.

109. Shew that if a triangle inscribed in a circle be isosceles, having each of its sides double the base, the squares described upon the radius of the circle and one of the sides of the triangle, shall be to each other in the ratio of 4: 15.

110. APB is a quadrant, SPT a straight line touching it at P, PM perpendicular to CA; prove that triangle SCT: triangle ACB:: triangle ACB: triangle CMP.

111. If through any point in the arc of a quadrant whose radius is R, two circles be drawn touching the bounding radii of the quadrant, and r, r' be the radii of these circles: shew that '= R.

112. If R be the radius of the circle inscribed in a right-angled triangle ABC, right-angled at 4; and a perpendicular be let fall from A on the hypotenuse BC, and if r, r be the radii of the circles inscribed in the triangles ADB, ACD: prove that r2 + p12 = R3.

XIV.

113. If in a given equilateral and equiangular hexagon another be inscribed, to determine its ratio to the given one.

114. A regular hexagon inscribed in a circle is a mean proportional between an inscribed and circumscribed equilateral triangle.

115. The area of the inscribed pentagon, is to the area of the circumscribing pentagon, as the square on the radius of the circle inscribed within the greater pentagon, is to the square on the radius of the circle circumscribing it.

116. The diameter of a circle is a mean proportional between the sides of an equilateral triangle and hexagon which are described about that circle.

BOOK XI.

DEFINITIONS.

I.

A SOLID is that which hath length, breadth, and thickness.

II.

That which bounds a solid is a superficies.

III.

A straight line is perpendicular, or at right angles to a plane, when it makes right angles with every straight line meeting it in that plane.

IV.

A plane is perpendicular to a plane, when the straight lines drawn in one of the planes perpendicular to the common section of the two planes, are perpendicular to the other plane.

V.

The inclination of a straight line to a plane, is the acute angle contained by that straight line, and another drawn from the point in which the first line meets the plane, to the point in which a perpendicular to the plane drawn from any point of the first line above the plane, meets the same plane.

VI.

The inclination of a plane to a plane, is the acute angle contained by two straight lines drawn from any the same point of their common section at right angles to it, one upon one plane, and the other upon the other plane.

VII.

Two planes are said to have the same, or a like inclination to one another, which two other planes have, when the said angles of inclination are equal to one another.

VIII.

Parallel planes are such as do not meet one another though produced.

IX.

A solid angle is that which is made by the meeting, in one point, of more than two plane angles, which are not in the same plane.

X.

Equal and similar solid figures are such as are contained by similar planes equal in number and magnitude.

XI.

Similar solid figures are such as have all their solid angles equal, each to each, and are contained by the same number of similar planes.

P