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hartnn Group Title

there are nine points on or inside the unit square.prove that u will always find three points, which consist a triangle with area not greater than (1/8)

  • one year ago
  • one year ago

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  1. hartnn Group Title
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    what i did: since the nine points need to be farthest from each other to get the max value of area, one possible configuration is shown in diagram. then it can easily be shown that the are of that 45-45-90 triangle is 1/2(1/2)(1/2) = 1/8 which is max. but i want some alternative general approach , if any. |dw:1352030299200:dw|

    • one year ago
  2. chihiroasleaf Group Title
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    should we choose non-collinear points? (the three points) can't we choose the triangle like below? or it will mean that I choose four points? :D |dw:1352030543732:dw|

    • one year ago
  3. hartnn Group Title
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    the triangle formed should not contain any other points on r inside it.

    • one year ago
  4. chihiroasleaf Group Title
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    it's said that the points randomly marked on or inside the square, can I put the points like this? :D |dw:1352030942810:dw|

    • one year ago
  5. hartnn Group Title
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    i am sorry! the exact question is 'there are nine points on or inside the unit square.prove that u will always find three points, which consist a triangle with area not greater than (1/8)' and is edited now......

    • one year ago
  6. hartnn Group Title
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    does this change anything ?

    • one year ago
  7. sara12345 Group Title
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    it does. it gets easy now as now we can simply apply pigeonhole principle

    • one year ago
  8. hartnn Group Title
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    idk what is pigeonhole principle ?

    • one year ago
  9. sara12345 Group Title
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    divide the unit square into 8 equal triangles first

    • one year ago
  10. sara12345 Group Title
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    |dw:1352031901246:dw|

    • one year ago
  11. sara12345 Group Title
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    think of them as pigion holes. 8 pigeon holes

    • one year ago
  12. sara12345 Group Title
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    we need to assign 9 points to 8 triangles. the points are pogeons

    • one year ago
  13. hartnn Group Title
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    interesting!

    • one year ago
  14. sara12345 Group Title
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    9 pigeons, 8 holes. by pigeonhole principle, one pigeon must go into an already occupied hole

    • one year ago
  15. sara12345 Group Title
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    that proves, atleast one triangle will have area less than 1/8 becoz, each triangle area is 1/8

    • one year ago
  16. sara12345 Group Title
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    wiki has very simple and good explanaiton of pigeonhole principle http://en.wikipedia.org/wiki/Pigeonhole_principle

    • one year ago
  17. hartnn Group Title
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    nice!

    • one year ago
  18. chihiroasleaf Group Title
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    whoaa..., I never thought that this problem can be solved using pigeonhole principle :D

    • one year ago
  19. sara12345 Group Title
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    also implicitly weve used below to conclude fully : For a given square with any 3 points inside making a triangle, the maximum area of triangle is half the area of square

    • one year ago
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