Thread: Midpoint question

Let (xi,yi), i = 1,2,3,4,5 be a set of five distinct points with integer coordinates on the xy plane. Show that the midpoint of the line joining at least one pair of the points has integer coordinates.

I understand how to graph this, but I am unsure on how to graph the midpoint to show this. Any help would be very appreciated!

Here are some hints.
We have four types of pairs: (e,e), (o,o), (e,o) and (o,e) (where ‘o’ stands for odd and ‘e’ stands for even).
Adding two odd numbers gives an even number as does adding two evens.
Midpoints are averages (i.e. divide by 2).
Put five pigeons into four holes.

So would it be suffcient enough to show that the midpoints of (1,1) and (5,5) have a midpoint of (3,3)?

Originally Posted by vexiked

So would it be suffcient enough to show that the midpoints of (1,1) and (5,5) have a midpoint of (3,3)?

no. you cannot choose specific examples like that. we have no idea if we have if those points are in our given list.

try to use Plato's suggestion. apply the pigeonhole principle, bearing in mind what he said