Math Is Fun Forum

atreyyu

Member

Registered: 2008-08-01

Posts: 5

Points 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?

Help!

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Points on a plane

There are two cases.

Case 1: There are no horizontally or vertically adjacent colours. This means that the points are alternately red and blue, so diagonal points are the same colour. Hence, the points (1,1) and (1,−1) are the same colour as the origin (0,0) and they form a right isosceles triangle.

Case 2: There is a pair horizontally or vertically adjacent colours. Without loss of generality, suppose these two points are (0,0) and (1,0). (The analysis is similar if the points are vertically adjacent.) If any one of (0,1) and (0,−1) is of the same colour, we would have a right isosceles triangle by joining it to (0,0) and (1,0). Otherwise, those two points are of opposite colour to (0,0) and (1,0). Then look at (2,1) and (2,−1). If they’re both the same colour as (1,1), join them to (1,1); if one of them is different, then join that point to (0,1) and (0,−1).