8 Points in Space
Is it possible to place 8 points in , such that any 3 points determine an isosceles or equilateral triangle?
Note: For this problem, the triangle is allowed to be degenerate, meaning we have 3 points on a line.
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1 solution
Take the origin, the 5 vertices of the regular pentagon that lies on x 2 + y 2 = 1 , and the 2 points ( 0 , 0 , ± 1 ) .
Claim:
1. We can verify that this set satisfies the conditions.
2. Every set that satisfies the conditions is isomorphic to this set (by scaling, rotating and translating)
3. There is no 9 points that satisfies the conditions.
This is known as Isosceles Sets in the literature. Look it up if you're interested.