Dem.—Join AD. Describe a circle about the triangle ADI, cutting the lines AF, CD produced, if necessary, in K and L. Join IK, KL, LI. Now, the angles KLG, FCG are each [III. xxi.] equal to the angle GAD. Hence they are equal. Therefore KL is parallel to CF. Similarly, LI is parallel to CH, and KI to FH; hence the triangles KLI, FCH are homothetic. Hence the lines joining corresponding vertices are concurrent. Therefore the points I, H, G are collinear. 73. If two sides of a triangle circumscribed to a given circle be given in position, but the third side variable, the circle described about the triangle touches a fixed circle. 74. If two sides of a triangle be given in position, and if the area be given in magnitude, two points can be found, at each of which the base subtends a constant angle. 75. If a, b, c, d denote the sides of a cyclic quadrilateral, and s its semiperimeter, prove its area = . 76. If three concurrent lines from the angles of a triangle ABC meet the opposite side in the points A′, B′, C′, and the points A′, B′, C′ be joined, forming a second triangle A′B′C′, 77. In the same case the diameter of the circle circumscribed about the triangle ABC = AB′.BC′.CA′ divided by the area of A′B′C′. 78. If a quadrilateral be inscribed in one circle, and circumscribed to another, the square of its area is equal to the product of its four sides. 79. If on the sides AB, AC of a triangle ABC we take two points D, E, and on their line of connexion F, such that prove the triangle BFC = 2ADE. 80. If through the middle points of each of the two diagonals of a quadrilateral we draw a parallel to the other, the lines drawn from their points of intersection to the middle points of the sides divide the quadrilateral into four equal parts. 81. CE, DF are perpendiculars to the diameter of a semicircle, and two circles are described touching CE, DE, and the semicircle, one internally and the other externally; the rectangle contained by the perpendiculars from their centres on AB is equal to CE.DF. 82. If lines be drawn from any point in the circumference of a circle to the angular points of any inscribed regular polygon of an odd number of sides, the sums of the alternate lines are equal. 83. If at the extremities of a chord drawn through a given point within a given circle tangents be drawn, the sum of the reciprocals of the perpendiculars from the point upon the tangents is constant. 84. If a cyclic quadrilateral be such that three of its sides pass through three fixed collinear points, the fourth side passes through a fourth fixed point, collinear with the three given ones. 85. If all the sides of a polygon be parallel to given lines, and if the loci of all the angles but one be right lines, the locus of the remaining angle is also a right line. 86. If the vertical angle and the bisector of the vertical angle be given, the sum of the reciprocals of the containing sides is constant.