Question:
If the length of the sides BC, CA and AB of a △ABC are a, b and c respectively and AD is the bisector of ∠A then find the length of BD and DC.
Solution:
Let DC = x
∴ BD = a − x
By using angle bisector theore in △ABC, we have
$\frac{\mathrm{AB}}{\mathrm{AC}}=\frac{\mathrm{BD}}{\mathrm{DC}}$
$\Rightarrow \frac{c}{b}=\frac{a-x}{x}$
$\Rightarrow c x=a b-b x$
$\Rightarrow x(b+c)=a b$
$\Rightarrow x=\frac{a b}{(b+c)}$
Now,
$a-x=a-\frac{a b}{b+c}$
$=\frac{a b+a c-a b}{b+c}$
$=\frac{a c}{(a+b)}$