Plot five points at random at the intersections of a coordinate grid. Between each pair of points a line segment can be drawn.
Prove that the midpoint of at least one of these segments occurs at an intersection of grid lines.
With 5 points there must be either 3 all with even x coordinates or 3 all with odd x coordinates. Choose the three points that fit this criterion.
Of the three chosen, at least two must have the same y-coordinate parity. These two points have the same y-coordinate parity and therefore differ by an even number in their y coordinates, and also have the same true of their x coordinates. Therefore taking the mean in either dimension results in an integer, for the midpoint.
Posted by Charlie
on 2016-05-18 19:49:08