Step 4 seems a bit glib... :) It would follow straightforwardly if we knew that the largest rectangle in a circle is a square. The easiest way I can think of

Step 4 seems a bit glib... :) It would follow straightforwardly if we knew that the largest rectangle in a circle is a square. The easiest way I can think of right now to see this is by Lagrange optimization (which we could also apply just as well directly to the 3d problem): the area of a rectangle of width and height <x, y> has a gradient of <y, x>, which can only be normal to the circle (which has normal vector <x, y>) when x = y. Another way is to think of <x, y> as proportional to <cos(t), sin(t)>; the area is then cos(t) * sin(t), which is proportional to sin(2t), and thus clearly maximized when x = y.