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hartnn

  • 2 years ago

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)

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  1. hartnn
    • 2 years ago
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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|

  2. chihiroasleaf
    • 2 years ago
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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|

  3. hartnn
    • 2 years ago
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    the triangle formed should not contain any other points on r inside it.

  4. chihiroasleaf
    • 2 years ago
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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|

  5. hartnn
    • 2 years ago
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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......

  6. hartnn
    • 2 years ago
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    does this change anything ?

  7. sara12345
    • 2 years ago
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    it does. it gets easy now as now we can simply apply pigeonhole principle

  8. hartnn
    • 2 years ago
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    idk what is pigeonhole principle ?

  9. sara12345
    • 2 years ago
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    divide the unit square into 8 equal triangles first

  10. sara12345
    • 2 years ago
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    |dw:1352031901246:dw|

  11. sara12345
    • 2 years ago
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    think of them as pigion holes. 8 pigeon holes

  12. sara12345
    • 2 years ago
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    we need to assign 9 points to 8 triangles. the points are pogeons

  13. hartnn
    • 2 years ago
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    interesting!

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

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

  16. sara12345
    • 2 years ago
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    wiki has very simple and good explanaiton of pigeonhole principle http://en.wikipedia.org/wiki/Pigeonhole_principle

  17. hartnn
    • 2 years ago
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    nice!

  18. chihiroasleaf
    • 2 years ago
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    whoaa..., I never thought that this problem can be solved using pigeonhole principle :D

  19. sara12345
    • 2 years ago
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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

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