Scalene triangles and an equilateral triangle inscribed with the same incenter

Lorian Saceanu (AoPS)

If the scalene triangle ABC and the equilateral triangle LMN have the same incenter, the points L, M, N are on the sides (AB), (AC), (BC) , then ABC is unique (?).

Observation: We will not consider the trivial case when triangle ABC is equilateral too. This is why I specified that ABC is a scalene triangle.