The following problems concern a rigid four-legged table whose feet may be considered as coplanar points, on a floor to be treated as rigid, conveniently smooth and large, but not flat.

1. (a) If the feet form a square, show that there is a position where all feet are on the floor.

2. (b) As (a), but with isosceles trapezium in place of square.

Of course (b) includes (a), but the diminished symmetry calls for a subtler argument.

1. (c) If the feet do not form a cyclic quadrilateral, show that there may be no position with all feet on the floor.