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I think the question might be better put this way:

Pick 5 random points on an infinite plane. Draw lines between all of them. One of those lines will contain another integer point. Prove this is true for every set of 5 random points.

You chose 6 points, and there are 3 line segments among them containing (1,1).

The line segment described by (0,0) (2,2) goes through the integer point (1,1).