In $$\triangle$$ABC, AD,the bisector of $$\angle$$A, meets BC at D. If BC = a, AC = b and AB =c, then BD - DC =

According to angle bisector theorem

$$\frac{AB}{AC}=\frac{BD}{DC}$$

$$\frac{c}{b}=\frac{BD}{DC}$$

Add 1 in both side

$$\frac{c}{b}+1=\frac{BD}{DC}+1$$

$$\frac{c+b}{b}=\frac{BD+DC}{DC}$$

$$\frac{c+b}{b}=\frac{a}{DC}$$

$$DC=\frac{ab}{b+c}$$ ------I

$$\frac{b}{c}=\frac{DC}{BC}$$

Add 1 in both side

$$\frac{b}{c}+1=\frac{DC}{DB}+1$$

$$\frac{c+b}{c}=\frac{DC+DB}{DB}$$

$$\frac{c+b}{c}=\frac{a}{DB}$$

$$DB=\frac{ac}{b+c}$$ ------ii

$$BD-DC=\frac{ac}{b+c}-\frac{ab}{b+c}=\frac{a\left(c-b\right)}{b+c}$$