The midpoint of the line joining two grid points (x₁,y₁) and (x₂,y₂) is located at ((x₁ + x₂)/2,(y₁ + y₂)/2). The latter will be a grid point iff its coordinates are integers. The x-coordinate will be integer iff x₁ and x₂ have the same parity, i.e., iff they are either both even or both odd. Out of 5 points, at least three satisify this condition. But the same is true of the y-coordinate. And out of the selected three points, at least two have y-coordinate with the same parity.

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References

1. A. Engel, Problem-Solving Strategies, Springer Verlag, 1998, pp. 61-62