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    Show a pair has a midpoint with integer coordinates

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    Let p = {(x1, y1), (x2, y2), (x3, y3), (x4, y4), (x5, y5)} be a set of five distinct points in the plane , each of which has integer coordinates. Show that some pair has a midpoint that has integer coordinates.

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    Solution Preview

    In order for a pair to have integer co ordinates , the sum of the pair should be even ( mid point = (x1 +x2)/2.

    Now : even + even = even
    odd + odd = even

    Solution Summary

    The expert shows a pair on the midpoint with integer coordinates. A set of five distinct points in the plane are determined.