<Tri-2025-2185>, hint

Suppose incircle of ABC  touches BC, CA, AB at D,E,F respectively

Consider trianges IDP, IEN, IFM which are congruent, then IM=IN=IP, which

means I is the circumcenter O of triangle MNP. Moreover  if O(=I) is the mass center G 

of MNP then MNP be equilateral. Finally triangle ABC be equilateral.

OG=(OA+OB+OC)/3, if OA+OB+OC=0  thenABC is equilateral