<Tri-2205-2231>, hint

Consider Vector sysyem.

Let O be the circumcenter of ABC and vectors OA,OB,OC=be a,b,c  respectively. and AM/MB=k/1-k, MR/RN=1-k/k . 

Then M=kb+(1-k)a, N=kc+(1-k)b, P=ka+(1-k)c

R=(1-k)N+kM. S=(1-k)P+kN, T(1-k)M+kP, AB=a-b, TR=(3^2k-3k+1)(a-b)

Hence AB//TR, similarly BC//RS, CA//ST. Triangles ABC and TRS are simialr

From 3^2-3k+1, we find  when k=1/2 TR is minimum.