So as I understand what you would suggest:
Let A,B,C be the 3 points of the main triangle.
Let a,b,c be the length of segments BC,AC,AB.
Let O be the center point we are looking for such as angles OAB=OAC=OBC=120deg.
Let d, e, f be the lengths of OA,OB,OC
As cos(O)=-1/2 per definition whatever...