<Tri-2025-2240>, hint 

Let circumcenter of ABC be O,  vectors OA,OB,OC be a,b,c respectively.

DB/DC=EC/EA=FA/FB=t/(1-t), area of ABC be S and area of BMC, CNA, APB=S'

OD=tc+(1-t)b, AD=OD-OA=[tc+(1-t)b-a], AM=AD x(S+S')/S=[tc+(1-t)b-a](1+k), k=S'/S

OM=OA+AM=[tc+(1-t)b-a](1+k)+a

Similarly  ON=[ta+(1-t)c-b](1+k)+b, OP=[tb+(1-t)a-c](1+k)+c

OM+ON+OC=a+b+c, which is centroid of ABC