Let Q be the cube with the set of vertices {(x1, x2, x3) ∈ R3: x1, x2, x3 ∈ {0,1}}. Let F be the set of all twelve lines containing the diagonals of the six faces of cube Q. Let S be the set of all four lines containing the main diagonals of the cube Q; for instance, the line passing through the vertices (0,0,0) and (1,1,1) is in S. For lines l1 and l2, let d(l1,l2) denote the shortest distance between them. Then the maximum value of d(l1,l2) as l1 varies over f and l2 varies over S, is
Consider an obtuse-angled triangle ABC in which the difference between the largest and the smallest angle is \(\frac{\pi}{2}\) and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.Then the inradius of the triangle ABC is
Consider an obtuse-angled triangle ABC in which the difference between the largest and the smallest angle is \(\frac{\pi}{2}\) and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.Let a be the area of the triangle ABC. Then the value of (64a)2 is