Take the G point on AC and make AG=AE.

Connecting GF

It is proved that △AEF is equal to △AGF.

Because AD bisects angle BAC, angle BAD= angle DAC.

AG=AE, AF=AF, so △AEF is equal to △AGF(SAS).

So FE=FG angle EFA= angle GFA

It is proved again that △CFG is all equal to △CFD.

Angle DCF= Angle GCF because CE bisects angle BCA.

Because angle EFA= angle GFA angle EFA= angle DFC (equal to the vertex angle)

So angle DFC= angle GFC

CF=CF

So △CFG is all equal to △ △CFD (ASA).

So GF=DF

And because it is known that FE=FG

So FE=FD