Four (dancing) points

Move the red point and observe, what is going on with the other three points.[br]Click 'Show vectors', observe and think...[br]How could you describe green, blue and brown point in terms of the red point

Fixed Point Theorem

Place an identical but scaled down map over the big original.[br]Is there a point where the same place on both maps coincide?[br]You can move and rotate (and scale) the small map...[br]Can you position the small map so that the answer to the above question would be affirmative?[br]Could there be more than one such point? Why ... ?[br]Choosing any positioning of the small map, is there a point where the same place on both maps coincide ?[br]Can you prove or disprove it?[br]Explore the applet by appropriate clicks... and think ...[br]Can you now answer the above questions?
You can simplify the problem by thinking about regular 'x' and 'y' coordinates (longitude and latitude) on a square and its sized down copy.[br]Of course, square (square map) could be changed to rectangular...