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 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)
 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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hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3what 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

chihiroasleaf
 2 years ago
Best ResponseYou've already chosen the best response.0should 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

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3the triangle formed should not contain any other points on r inside it.

chihiroasleaf
 2 years ago
Best ResponseYou've already chosen the best response.0it's said that the points randomly marked on or inside the square, can I put the points like this? :D dw:1352030942810:dw

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3i 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
 2 years ago
Best ResponseYou've already chosen the best response.3does this change anything ?

sara12345
 2 years ago
Best ResponseYou've already chosen the best response.2it does. it gets easy now as now we can simply apply pigeonhole principle

hartnn
 2 years ago
Best ResponseYou've already chosen the best response.3idk what is pigeonhole principle ?

sara12345
 2 years ago
Best ResponseYou've already chosen the best response.2divide the unit square into 8 equal triangles first

sara12345
 2 years ago
Best ResponseYou've already chosen the best response.2dw:1352031901246:dw

sara12345
 2 years ago
Best ResponseYou've already chosen the best response.2think of them as pigion holes. 8 pigeon holes

sara12345
 2 years ago
Best ResponseYou've already chosen the best response.2we need to assign 9 points to 8 triangles. the points are pogeons

sara12345
 2 years ago
Best ResponseYou've already chosen the best response.29 pigeons, 8 holes. by pigeonhole principle, one pigeon must go into an already occupied hole

sara12345
 2 years ago
Best ResponseYou've already chosen the best response.2that proves, atleast one triangle will have area less than 1/8 becoz, each triangle area is 1/8

sara12345
 2 years ago
Best ResponseYou've already chosen the best response.2wiki has very simple and good explanaiton of pigeonhole principle http://en.wikipedia.org/wiki/Pigeonhole_principle

chihiroasleaf
 2 years ago
Best ResponseYou've already chosen the best response.0whoaa..., I never thought that this problem can be solved using pigeonhole principle :D

sara12345
 2 years ago
Best ResponseYou've already chosen the best response.2also 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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