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.