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Math Help - Squares and Pigeonhole Principle

  1. #1
    Senior Member I-Think's Avatar
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    Squares and Pigeonhole Principle

    Let S be a square region of side length 2. Show that among any 9 points in the square, there are 3 which form a triangle of area \leq{\frac{1}{2}.

    I subdivided S into 8 triangles of equal area, \frac{1}{2}.
    Now I divide into 2 cases
    Case A: Where 3 points are in one triangle.
    Then the area of said triangle \leq{\frac{1}{2}.

    Case B: A max of 2 points are in any triangle
    I'm having trouble with this case. Help please?
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  2. #2
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    Re: Squares and Pigeonhole Principle

    Quote Originally Posted by I-Think View Post
    Let S be a square region of side length 2. Show that among any 9 points in the square, there are 3 which form a triangle of area \leq{\frac{1}{2}.
    Consider a unit square, a square of side length one and area of one square unit.
    Any three points from that unit square that do form a triangle,
    The area of such a triangle is \le0.5 square units.
    Can you show that?

    Now divide your given square into four sub-squares each of one square unit each.

    Can you finish?
    Last edited by Plato; July 7th 2013 at 12:57 PM.
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