Two Colors - All Distances

Points in the plane are each colored with one of two colors: red or blue. The set of distances between the blue points is blue and the set of distances between the red points is red. Prove that either one or the other contains all the positive reals.

Solution

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Copyright © 1996-2018 Alexander Bogomolny

Points in the plane are each colored with one of two colors: red or blue. The set of distances between the blue points is blue and the set of distances between the red points is red. Prove that either one or the other contains all the positive reals.

Solution

Assume to the contrary that the blue set lacks distance b and the red set lacks distance r. Choose a blue point B and form an isosceles triangle ABC with AB = BC = b and AC = r. Then either both A and C are red or one of them is blue. If they are both red we have two red points at distance r. If one of them blue, we have two blue points at distance b.

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|Contact| |Front page| |Contents| |Coloring Plane|

Copyright © 1996-2018 Alexander Bogomolny

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