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.
87. If P, P′ denote the areas of two regular polygons of any common number of sides, inscribed and circumscribed to a circle, and Π, Π′ the areas of the corresponding polygons of double the number of sides; prove Π is a geometric mean between P and P′, and Π′ a harmonic mean between Π and P′.
88. The difference of the areas of the triangles formed by joining the centres of the circles described about the equilateral triangles constructed—(1) outwards; (2) inwards—on the sides of any triangle, is equal to the area of that triangle.