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5 points (last one i swear)

  1. Jul 1, 2007 #1
    im so embarrased askin so much :blushing:

    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.
     
  2. jcsd
  3. Jul 1, 2007 #2
    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.
     
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