<Tri-2024-2119>, hint

Reflect  B wrt AC into B', and reflect A wrt BC into A'

Then <A'CB'=3 x 60=180. Let  C' be the intersection of lines B'A with A'B,

 Then triangles ABC and A'B'C' are similar. A'B'=AC+BC.

let midpoint s of AB' and BA" be R' and R". segment R'R" cut AC in Q and BC in P

R'R" is minimum perimeter of triangle PQR.