hartnn
  • hartnn
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)
Geometry
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chestercat
  • chestercat
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hartnn
  • hartnn
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|
chihiroasleaf
  • chihiroasleaf
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|
hartnn
  • hartnn
the triangle formed should not contain any other points on r inside it.

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chihiroasleaf
  • chihiroasleaf
it's said that the points randomly marked on or inside the square, can I put the points like this? :D |dw:1352030942810:dw|
hartnn
  • hartnn
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......
hartnn
  • hartnn
does this change anything ?
anonymous
  • anonymous
it does. it gets easy now as now we can simply apply pigeonhole principle
hartnn
  • hartnn
idk what is pigeonhole principle ?
anonymous
  • anonymous
divide the unit square into 8 equal triangles first
anonymous
  • anonymous
|dw:1352031901246:dw|
anonymous
  • anonymous
think of them as pigion holes. 8 pigeon holes
anonymous
  • anonymous
we need to assign 9 points to 8 triangles. the points are pogeons
hartnn
  • hartnn
interesting!
anonymous
  • anonymous
9 pigeons, 8 holes. by pigeonhole principle, one pigeon must go into an already occupied hole
anonymous
  • anonymous
that proves, atleast one triangle will have area less than 1/8 becoz, each triangle area is 1/8
anonymous
  • anonymous
wiki has very simple and good explanaiton of pigeonhole principle http://en.wikipedia.org/wiki/Pigeonhole_principle
hartnn
  • hartnn
nice!
chihiroasleaf
  • chihiroasleaf
whoaa..., I never thought that this problem can be solved using pigeonhole principle :D
anonymous
  • anonymous
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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