* Proof:* Take * f* to be any continuous function on a circle. Draw a line through the center of the circle, intersecting it at two opposite points * A* and * B* . Let * d* be defined by the difference * f* (* A* ) − * f* (* B* ). If the line is rotated 180 degrees, the value −* d* will be obtained instead. Due to the intermediate value theorem there must be some intermediate rotation angle for which * d* = 0, and as a consequence * f* (* A* ) = * f* (* B* ) at this angle.