CGMO 2003 Problem 7

Let the sides of a scalene triangle $\triangle ABC$ be $AB = c,$ $BC = a,$ $CA =b,$ and $D, E , F$ be points on $BC, CA, AB$ such that $AD, BE, CF$ are angle bisectors of the triangle, respectively. Assume that $DE = DF.$ Prove that

(1) $\frac{a}{b+c} = \frac{b}{c+a} + \frac{c}{a+b}$

(2) $\angle BAC > 90^{\circ}.$