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I've just been thinking about for what values of n we can place n points in the plane so that any three of those points define an isosceles triangle. A triangle, square and pentagon work for 3,4 and 5, and to get 6 just place a point in the centre of the pentagon.

The question is can we do this for 7 points in the plane?

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This was a problem in the American Mathematical Monthly in 1946, proposed by Erdős. If you have access to JSTOR, here's a solution from 1947: The solution is by L.M. Kelly: – Jonas Meyer Dec 15 '10 at 1:21

Here is a recent paper related to this topic:

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