Answer
--Showing that the midpoint of at least one pair of given points has
integer coordinates.
Work Step by Step
-The midpoint of the segment joining the points (a, b, c) and (d, e, f )
is ((a+d)/2, (b+e)/2, (c+f )/2).
-It has integer coefficients
-if and only if a and d have the same parity, b and e have the same parity, and c and f have the same parity.
-Because there are eight possible triples of parity [such as (even, odd, even)],
--by the pigeonhole principle:
- at least two of the nine points have the same triple of parities.
-The midpoint of the segment joining two such points has integer coefficients.
