5 points (last one i swear)

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SUMMARY

The discussion centers on proving that given five distinct lattice points in the plane, there exists a line segment between two of them that contains another lattice point in its interior. Utilizing the Pigeonhole Principle, the argument establishes that among the four parity combinations (even, even), (even, odd), (odd, even), and (odd, odd), at least two points must share the same parity. By applying the midpoint formula, it is demonstrated that the midpoint of these two points is also a lattice point, confirming the existence of an interior lattice point.

PREREQUISITES
  • Understanding of lattice points and integer coordinates
  • Familiarity with the Pigeonhole Principle
  • Knowledge of midpoint formula in coordinate geometry
  • Basic concepts of parity (even and odd numbers)
NEXT STEPS
  • Study the Pigeonhole Principle in combinatorial mathematics
  • Explore properties of lattice points in geometry
  • Learn about coordinate geometry and the midpoint formula
  • Investigate parity and its applications in number theory
USEFUL FOR

This discussion is beneficial for mathematicians, educators, and students interested in combinatorial geometry, particularly those studying properties of lattice points and their relationships in the coordinate plane.

barbiemathgurl
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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.
 
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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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