A rectangle is inscribed within another rectangle of dimensions 6 units by 8 units. The rectangle has been inscribed in it with its vertices on the sides of the rectangle in such a way that if the smaller rectangle is rotated slightly about the centre it still is confined within the boundary of the larger rectangle. How can I find the minimum possible perimeter of such a rectangle? I am interested in an intuitive proof by simple geometry and not by coordinate geometry.
Intuitively, I think you are asking for the smallest circle that will fit into the 6x8 rectangle, and still touch all 4 sides. Your rectangle is then the corresponding touch points that actually form a rectangle. Again, intuitively (without calculation) this seems to be the minimum perimeter. I think the calculation might be provided by Jaap in a comment.