The set Z times Z consisting of those points in the plane wi

The set Z times Z, consisting of those points in the plane with both coordinates being integers, is called the set of lattice points in the plane. Show that given any five lattice points in the plane, there exists a pair determining a segment whose midpoint is also a lattice point in the plane.

Solution

SOLUTION :-

Let E represent the even integers and O represent the odd integers.

There are four combinations of integers

(E,O) ,(O,E), (E,E) , (O,O)

we know that the mid points of two ordered pairs of same form yields another ordered pair of that form.

And we also know that the midpoints of two ordered pairs of different forms will not yield integral coordinates.

There are four categories of different combinations of ordered integral ordered pairs and five lattice points. so one category must have at least two objects.

The ordered pair of the same form and so the line segment that they produce creates a segment with a midpoint having a integral coordinates.