<Qua-2025-771>, hint

 D is the excenter of triangle EBF,

Coordinate system ;

B(0,0)  E(0, 1-t), C(1,0), D(1,1)  DE; y-1=t(x-1), AC=y=-x+1, Then P(t/(1+t), 1/(1+t))

PF ; y=-1/t x+2/(1+t), and calculations show EF=AE+FC.