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# Every point in the plane can be expressed in terms of its x- and y-coordinates. A point whose coordinate are both integers is called a lattice point ....

Every point in the plane can be expressed in terms of its x- and y-coordinates. A point whose coordinate are both integers is called a lattice point. For example, the points (1, 2), ( 3, 8) and the origin (0, 0) are all lattice points, but (1.5, 0) is not.

For example, in the diagram below, the five lattice points are (1, 0), (5, 5), (6, 3), (9, 6), (10, 2), and the line

segment determined by the two points (1, 0) and (9, 6) has the lattice point (5, 3) as midpoint.

Prove the following fact using the pigeonhole principle:Given five distinct lattice points in the plane, at least one of the line segments determined by these points has a lattice point as its midpoint.