# Thread: Triangles on a plane

1. ## Triangles on a plane

We have a set of points on a plane, and each point has been coloured either red or blue. How to prove that there exists a right and isosceles triangle, whose vertices are all the same colour?

2. Originally Posted by atreyyu
We have a set of points on a plane, and each point has been coloured either red or blue. How to prove that there exists a right and isosceles triangle, whose vertices are all the same colour?
You can't, at least not with that information. How many points are there? How many are red? How are they positioned?

You'll have to give us the entire problem.

3. This is the entire problem. As a point has no width or height, let's say there is an infinite number of them. Take a table of size 3x3 and try to insert red or blue points so that no three of them form a right isosceles triangle. You can't do that, and this is what I have to prove.

4. Originally Posted by atreyyu
As a point has no width or height, let's say there is an infinite number of them.
If there is an infinite number of them than any possible shape will be made (that's less than a 3x3 box) So a right isosceles triangle will inevitably occur.

It is like the monkey's on a keyboard. Given an infinite amount of time, they will write the completed works of Shakespeare.

5. Originally Posted by Quick
If there is an infinite number of them than any possible shape will be made (that's less than a 3x3 box) So a right isosceles triangle will inevitably occur.

It is like the monkey's on a keyboard. Given an infinite amount of time, they will write the completed works of Shakespeare.
O, when she's angry, she is keen and shrewd!
She was a vixen when she went to school;
"And though she be but little, she is fierce."

~A Midsummer Night's Dream ~ W. Shakespeare