Three circles of radii $a,b,c$ touch one another externally and the tangents at their points of contact meet at a point $P$.Prove that distance of $P$ from either of the points of contact is $\sqrt{\frac{abc}{a+b+c}}$.

In this question,i tried but did not get near to the answer.Not much is given about the circles except radii.Please help me.


Firstly, let call the centers points of three circles $O_1$, $O_2$, $O_3$.

You can prove that $P$ is the center of the incircle of the triangle $\Delta O_1O_2O_3$.

Note that the sides of this triangle are $a+b$, $b+c$ and $c+a$.

By Heron's formula, the area of this triangle is $\Delta=\sqrt{(a+b+c)(abc)}$.

Finally, you have the distance of $P$ from the points of contact is the radius of the incircle of $\Delta O_1O_2O_3$, since:

$$ r = \frac{2\Delta}{(a+b)+(b+c)+(c+a)}.$$

Now, you can get the conclusion.


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