8 Points in Space

Geometry Level 4

Is it possible to place 8 points in R 3 \mathbb{R} ^3 , 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

Calvin Lin Staff
May 18, 2017

Take the origin, the 5 vertices of the regular pentagon that lies on x 2 + y 2 = 1 x^2 + y^2 = 1 , and the 2 points ( 0 , 0 , ± 1 ) (0, 0, \pm 1 ) .

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.

Pentagon, you mean :P

Shourya Pandey - 4 years ago

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Ah yes. Thanks.

Calvin Lin Staff - 4 years ago

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