<Polygon-2023-262>, hint

Let AF=an CD=d. AD and CF intersect at P, <PAF=p <PFA=q 

In triangle PAF, AF/sin60=PA/sinp,=PF/sinq

PA+PF=AF/sin60(sinp+sinq)=AF/ sin60 2sin(p+q)/2cosp-q)/2=2AF/sin60 sin60 cos(p-q)/2

2AF cos(p-q)/2<2AF, Equality holds when p=q=60.

Similarly in other 11 triangles, above logic applied.