Drepte perpendiculare in triunghiul cu A=120 grade (Anglia)

[mihai miculita]

Fie $ABC$ un triunghi in care $m(\widehat{BAC})=120^0.$ Notam cu $D,\,E$ si $F-$picioarele bisectoarelor interioare duse din varfurile $A,\,B$ si respectiv $C.$
Aratati ca: $DE\perp DF.$ (Problema 2/ONM-ANGLIA, runda a 2-a, din 1 feb.2005)

[sunken rock]

E is the B-excenter of $\triangle ABD$ and F is the C-excenter of $\triangle ADC$ (AC is the external bisector of $\angle BAD$, AB the external bisector of $\angle DAC$), hence DE, DF are bisectors of $\angle ADC$.

Best regards,
sunken rock