<Qua-2025-798>, hint

Let <ACD=X, <CAD=Y 

<AFC=X+Y( <ABC+<AFC=180)=<ADF, AD=AF, Similarly CE=CD

Suppose Angle bisectors of <DAF , <DCE meet at O'

Above angles bisectors are  also segments bisectors of  DF, DE , which

meet at O , the circumcenter of DEF.

<CO'A=X+Y(By angle chasing)=<MON  (M,N are the midpoints of DF, DE)

Hence O=O'.