TITLE: Do all figures have a "centre", equidistant from all vertices?
QUESTION [6 upvotes]: Can we prove or disprove that :
There exists for any given closed figure, a point which is equidistant from all of its vertices? 
Any closed figure means literally any closed figure?
I am gonna instinctively say no, but How!?

REPLY [16 votes]: This is clearly false. First choose three random non-collinear points. Then there is a unique circle that goes through these three points, and hence there is a unique point (the circle's center) that is equidistant from the three points. Now add any other point which does not lie on the perimeter of the circle.