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.