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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
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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hartnnBest ResponseYou've already chosen the best response.3
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 454590 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

chihiroasleafBest ResponseYou've already chosen the best response.0
should we choose noncollinear 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

hartnnBest ResponseYou've already chosen the best response.3
the triangle formed should not contain any other points on r inside it.
 one year ago

chihiroasleafBest ResponseYou've already chosen the best response.0
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

hartnnBest ResponseYou've already chosen the best response.3
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

hartnnBest ResponseYou've already chosen the best response.3
does this change anything ?
 one year ago

sara12345Best ResponseYou've already chosen the best response.2
it does. it gets easy now as now we can simply apply pigeonhole principle
 one year ago

hartnnBest ResponseYou've already chosen the best response.3
idk what is pigeonhole principle ?
 one year ago

sara12345Best ResponseYou've already chosen the best response.2
divide the unit square into 8 equal triangles first
 one year ago

sara12345Best ResponseYou've already chosen the best response.2
dw:1352031901246:dw
 one year ago

sara12345Best ResponseYou've already chosen the best response.2
think of them as pigion holes. 8 pigeon holes
 one year ago

sara12345Best ResponseYou've already chosen the best response.2
we need to assign 9 points to 8 triangles. the points are pogeons
 one year ago

sara12345Best ResponseYou've already chosen the best response.2
9 pigeons, 8 holes. by pigeonhole principle, one pigeon must go into an already occupied hole
 one year ago

sara12345Best ResponseYou've already chosen the best response.2
that proves, atleast one triangle will have area less than 1/8 becoz, each triangle area is 1/8
 one year ago

sara12345Best ResponseYou've already chosen the best response.2
wiki has very simple and good explanaiton of pigeonhole principle http://en.wikipedia.org/wiki/Pigeonhole_principle
 one year ago

chihiroasleafBest ResponseYou've already chosen the best response.0
whoaa..., I never thought that this problem can be solved using pigeonhole principle :D
 one year ago

sara12345Best ResponseYou've already chosen the best response.2
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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