Show, among 9 pts in rectangle [-1,1]x[0,1], there are 2 pts w/ dist. <1/sqrt[2]

[little_owl]

Please help me with this task
Show that, among 9 points in the rectangle R= { (x, y) ∈

² / -1≤x ≤1; 0≤ y≤1} there is
two points whose distance is less or equal to 1/√2

 

[Dr.Peterson]

Please help me with this task
Show that, among 9 points in the rectangle R= { (x, y) ∈

² / -1≤x ≤1; 0≤ y≤1} there is
two points whose distance is less or equal to 1/√2

This looks like a pigeonhole type problem. Are you familiar with that method?

Can you make 8 "pigeonholes" that a 9th "pigeon" won't fit into without sharing space?

 

[little_owl]

This looks like a pigeonhole type problem. Are you familiar with that method?

Can you make 8 "pigeonholes" that a 9th "pigeon" won't fit into without sharing space?

Unfortunately, I’m not familiar with method. Could you explain it? I would be very grateful.

 

[JeffM]

Please help me with this task
Show that, among 9 points in the rectangle R= { (x, y) ∈

² / -1≤x ≤1; 0≤ y≤1} there is
two points whose distance is less or equal to 1/√2

I seldom post on geometry problems because I have forgotten most theorems, but here is a method of attack that I suspect will work. It combines Dr. Peterson's pigeon holes and analytic geometry.

Start with the simpler problem of a unit square with corners at (0, 0) and (1, 1) containing 3 distinct points. The trivial case is that the points are colinear. The more difficult case is to show that if the points form a triangle, the length of at least one side is less than 1 / sqrt(2).

Now the geometry part is done. Your given triangle can be sub-divided into four unit squares. Those are your pigeon holes.

 

[Dr.Peterson]

Unfortunately, I’m not familiar with method. Could you explain it? I would be very grateful.

If you don't know about the pigeonhole principle (which you can look up), then can you tell us why you are working on this problem? If it is for a class, what methods do you know that might be used? If it is not for a class, what is its context?

Here are two sources (from a Google search): https://brilliant.org/wiki/pigeonhole-principle-definition/ ; https://www.math.hmc.edu/funfacts/ffiles/10001.4.shtml .