Question 171
Let $\triangle ABC$ be a triangle the lengths of whose sides $BC, CA, AB$, respectively, are denoted by $a, b, c$, respectively. Let the internal bisectors of the angles $\angle BAC, \angle ABC, \angle BCA$, respectively, meet the sides $BC, CA, AB$, respectively, at $D, E, F$, respectively. Denote the lengths of the line segments $AD, BE, CF$, respectively, by $d, e, f$, respectively. Prove that
$$
def = \frac{4abc(a+b+c)\Delta}{(a+b)(b+c)(c+a)},
$$
where $\Delta$ stands for the area of the triangle $ABC$.
$$
def = \frac{4abc(a+b+c)\Delta}{(a+b)(b+c)(c+a)},
$$
where $\Delta$ stands for the area of the triangle $ABC$.