TITLE: Collections of points containing only isosceles triangles
QUESTION [6 upvotes]: 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?

REPLY [2 votes]: Here is a recent paper related to this topic:
http://www.cims.nyu.edu/~pach/publications/isosceles.ps