As shown in the figure triangle ABC,AD is the bisector of angle BAC,E.F is the point on AB,AC respectively,and angle EDF+angle BAF=180 degree,prove DE=DF.
Make a vertical line from point D to AB and AC, and the points of intersection are M and N.
∵DN⊥AC, AD bisects ∠BAC
∴DM=DN
And ∠EDF+∠BAF=180
∴∠DEA+∠DFA=180
And ∠DEA+∠DEB=180
And ∠DEA+∠DEB=180
And ∠DFC+∠DEB=180
And ∠DEA+∠DEB=180
And ∠DEA+∠DEB=180
And ∠DFC=180 degrees. p>∴∠DFC=∠DEB
∴ΔDEM?±ΔDFN
So, DE=DF