5 points (last one i swear)

[barbiemathgurl]

im so embarrased askin so much

show that given 5 distinct lattice points in the plane (points with integer coordinates) there exists a line segment between both of them containing another lattice point on its interior.

 

[Kummer]

Consider parity.
In which case there are 4 possibilities:
(even,even);(even,odd);(odd,even);(odd,odd)

Now there are 5 points so by the Pigeonhole Principle two of them are of equal parity. Say (a,b) and (c,d).

By the midpoint formula we have that [(a+c)/2,(b+d)/2] is a midpoint of the line joining them. But since "a" and "c" have similar parity it means (a+c)/2 is an integer. Likewise, (b+d)/2 is an integer. That shows that this point lies on the interior of this line segment.