## 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)

1. 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|

2. 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|

3. hartnn

the triangle formed should not contain any other points on r inside it.

4. 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|

5. 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......

6. hartnn

does this change anything ?

7. sara12345

it does. it gets easy now as now we can simply apply pigeonhole principle

8. hartnn

idk what is pigeonhole principle ?

9. sara12345

divide the unit square into 8 equal triangles first

10. sara12345

|dw:1352031901246:dw|

11. sara12345

think of them as pigion holes. 8 pigeon holes

12. sara12345

we need to assign 9 points to 8 triangles. the points are pogeons

13. hartnn

interesting!

14. sara12345

9 pigeons, 8 holes. by pigeonhole principle, one pigeon must go into an already occupied hole

15. sara12345

that proves, atleast one triangle will have area less than 1/8 becoz, each triangle area is 1/8

16. sara12345

wiki has very simple and good explanaiton of pigeonhole principle http://en.wikipedia.org/wiki/Pigeonhole_principle

17. hartnn

nice!

18. chihiroasleaf

whoaa..., I never thought that this problem can be solved using pigeonhole principle :D

19. sara12345

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