<Cir-2024-762>, hint

Let centers of circles be (0,0), (d,0) A,A' be( r,o), (d+r',0)

Since AP and A'P' are parallel P(rcosQ, rsinQ) P'(d+r'cosQ, r' sinQ)

Hence Midpoint of PP', x=1/2(rcosQ+d+r'cosQ), y=1/2(rsinQ+r'sinQ)

Locus of midpoint of  PP' is a circle.