Question

Three circles, whose radii are $a,b,c$ touch each other externally and the tangents at their points of contact meet in a point. Then, the distance of this point from either of points of contact is

• A. $\sqrt{\frac{abc}{a+b+c}}$
• B. $\sqrt{\frac{abc}{a-b-c}}$
• C. $2\sqrt{\frac{abc}{a+b+c}}$
• D. None of these

Answer 1

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

Let $ID,IE,IF$ be the common tangents meeting at $I.$

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 $△ABC$ whose side are $b+c,c+a,a+b$

Now, $2s=b+c+c+a+a+b=2(a+b+c)$

$\therefore s=a+b+c$

$△=\sqrt{s(s-\overline{b+c})(s-\overline{c+a})(s-\overline{a+b})}=\sqrt{s\left(a\right)\left(b\right)\left(c\right)}=\sqrt{sabc}$

Now, $r=\frac{△}{s}=\frac{\sqrt{sabc}}{s}=\sqrt{\frac{abc}{s}}=\sqrt{\frac{abc}{a+b+c}}$