<Solid-2025-261 >, hint

Let  P on AB,  Q on BC, R on,CD, S on DA  such that PQRS is a rhombus.

  For PS//QR, PQ//SR,  AP/PB=x/(1-x), BQ/QC=/(1-y)/y, CR/RD=x/(1-x), DS/SA=(1-y)/y

  For PQ=QR=RS=SP, x^2+y^2-2xy cos60=PS^2, (1-x)^2+(1-y)^2-2(1-x)(1-y) cos60=PQ^2

 PS=PQ . Calculation shows  x+y=1. PR and QS should meet at  midpoint. 

which shows y/2=(1-y)/2 Finally x=y=1/2, and P,Q,R,S are midpoints of AB, BC, CD, DA