Red/Blue Points, Equilateral Triangle

Suppose every point in a plane is colored either red or blue. Show that there is an equilateral triangle somewhere in the plane whose vertices are all the same color.


Show/Hide Solution

Find two points, A and B, that are the same color, say blue. Then make a triangular array of points as shown.

  1. If C or D are blue, we’re done.
    So assume C is red, and D is red.
  2. If E is red, we’re done: △CDE.
    So assume E is blue.
  3. If F is blue, we’re done.
    So assume F is red.
  4. Now look at G.
    If G is blue, we have △AEG.
    If G is red, we have △CFG.


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