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. 89. In the same case, the sum of the squares of the sides of the two new triangles is equal to the sum of the squares of the sides of the original triangle. 90. If R, r denote the radii of the circumscribed and inscribed circles to a regular polygon of any number of sides, R′, r′, corresponding radii to a regular polygon of the same area, and double the number of sides; prove 91. If the altitude of a triangle be equal to its base, the sum of the distances of the orthocentre from the base and from the middle point of the base is equal to half the base. 92. 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 the sides is to four times the area. 93. Given the area of a parallelogram, one of its angles, and the difference between its diagonals; construct the parallelogram. 94. If a variable circle touch two equal circles, one internally and the other externally, and perpendiculars be let fall from its centre on the transverse tangents to these circles, the rectangle of the intercepts between the feet of these perpendiculars and the intersection of the tangents is constant. 95. Given the base of a triangle, the vertical angle, and the point in the base whose distance from the vertex is equal half the sum of the sides; construct the triangle. 96. If the middle point of the base BC of an isosceles triangle ABC be the centre of a circle touching the equal sides, prove that any variable tangent to the circle will cut the sides in points D, E, such that the rectangle BD.CE will be constant. 97. Inscribe in a given circle a trapezium, the sum of whose opposite parallel sides is given, and whose area is given. 98. Inscribe in a given circle a polygon all whose sides pass through given points. 99. If two circles X, Y be so related that a triangle may be inscribed in X and circumscribed about Y , an infinite number of such triangles can be constructed. 100. In the same case, the circle inscribed in the triangle formed by joining the points of contact on Y touches a given circle. 101. And the circle described about the triangle formed by drawing tangents to X, at the angular points of the inscribed triangle, touches a given circle.