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See diagram. I hope we can apply Sine and Cos trigonometric ratios here.

we are given DC = AB. angle B = 2C. AD is bisector of angle A.

AB / Sin C = AC / Sin B Sine of angles rule in ΔABC = AC / Sin 2C

=> AC / AB = 2 Sin C Cos C/ Sin C = 2 Cos C

But Cos C = CF / CD = CD / AB in triangle CFD. => AF = CF => AF = CF = AC / 2

now DF is the perpendicular bisector for the side AC.