BdMO Online Forum

In acute angled triangle $\Delta ABC$, considering a portion of side $BC$ as diameter a circle is drawn whose radius is $18$ units and it touches $AB$ and $AC$ side. Similarly, considering a portion of sides $AC$ and $AB$ as diameters, two other circles are drawn whose radii are $6$ and $9$ units respectively. What is the radius of the incircle of $\Delta ABC$?

Since we obviously have to work with areas and perimeters, we might as well try and figure out the area and perimeter of the triangle. Let $D$ be the centre of the circle with centre on $CB$. Note that by joining A with D, we get the area of the triangle $\Delta ABC$ to be $K=9(b+c)$. Continue this way, and finally conclude with $K=sr$.

rah4927 wrote:Let $D$ be the centre of the circle with centre on $AB$.

I think you wanted to tell that $D$ is the centre of the circle with centre on $BC$.
BTW,I have also solved this in the same way.Actually this is a very easy problem.I have posted this because the problem has an interesting property:Let $D$,$E$ and $F$ be the centers of the circles with center on $BC,CA,AB$,respectively.Then $AD,BE,CF$ are concurrent and they concur at the incenter of triangle $ABC$.Actually,they are the internal angle bisectors of angle $A,B,C$,respectively.The proof is quite easy.

tanmoy wrote:

rah4927 wrote:Let $D$ be the centre of the circle with centre on $AB$.

I think you wanted to tell that $D$ is the centre of the circle with centre on $BC$.

Yes, edited, thanks.