Question

Three circles with radii $a,b,c$ touch one another externally. If the tangents at contact points meet at point $I,$ then the distance of $I$ to the contact point of any two circles is:

• A. $\sqrt{\frac{a+b+c}{abc}}$
• B. $\frac{abc}{a+b+c}$
• C. $\sqrt{\frac{abc}{a+b+c}}$
• D. $\frac{a+b+c}{abc}$

Answer 1

The correct option is C $\sqrt{\frac{abc}{a+b+c}}$

Let $ID,IE,IF$ be the common tangents meeting at point $I$
Then $ID=IF,IE=IF$
$\therefore ID=IE=IF$
Since $ID\perp BC,IE\perp CA,IF\perp AB$
$\therefore I$ is the incentre of $\Delta ABC$ whose sides are $b+c,c+a,a+b$
Now, $2s=b+c+c+a+a+b=2(a+b+c)$
$\therefore s=a+b+c$
Area of triangle $\left(\Delta \right)=\sqrt{s[s-(b+c\left)\right][s-(c+a\left)\right][s-(a+b\left)\right]}$
$\Rightarrow \Delta =\sqrt{sabc}$

Now, required distance is $r=\frac{\Delta }{s}=\frac{\sqrt{sabc}}{s}=\frac{\sqrt{abc}}{\sqrt{s}}$
$r=\sqrt{\frac{abc}{a+b+c}}$