Question

If in a $\Delta ,R$ and $r$ are the circumradius and inradius respectively then find the H.M. of the ex-radii of the $\Delta .$

• A. The H.M. of the Ex-radii of $\Delta =3r$
• B. $4r$
• C. $r/2$
• D. $3r$

Answer 1

The correct option is A The H.M. of the Ex-radii of $\Delta =3r$

In $\Delta ABC$ whose ex-radii are $r_1,r_2,r_3$ respectively.
$\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=\frac{s-a+s-b+s-c}{\Delta }=\frac{s}{\Delta }$
$\Rightarrow \frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=\frac{1}{r}$
Now, HM of ex-radii $=\frac{3}{\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}}=3r$
$\therefore$ HM of ex-radii $=3r$