<Tri-2025-2220>, hint

Let C", B",A" be the midpoints of segments, through M parallel to AB,AC,BC.

Since <GC"M =90. GC"M is right triangle OG=OM=OC"

similarly OB"=OA"=OG =OC".

Since <GC"M=<GB"M=<GA"M=90, G,M,A",B",C" are concyclic and <C"MA"=60=<C"B"A"

A"B"C" is equilateral