algorithms – Find enclosing Equilateral triangle with minimum area

We have $$n$$ point in $$2D$$ space. We want to find a enclosing Equilateral triangle with minimum area in $$O(nlog n)$$.

My attempt:
I try to compute convex hull of points then i select each three points of hull vertex that first, 3 sides is equal and have minimum area, but the running time become to $$O(n^3)$$ because we need check combination 3 of $$O(n)$$ so it isn’t efficient.

Second try:
I think this problem have a relate with lover envelope, but i can’t formulate in dual manner.
any help to solve this problem in $$O(nlog n)$$ be appreciated.