Dhaka 2014,Secondary,P6
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Please don't post problems (by starting a topic) in the "X: Solved" forums. Those forums are only for showcasing the problems for the convenience of the users. You can always post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
Please don't post problems (by starting a topic) in the "X: Solved" forums. Those forums are only for showcasing the problems for the convenience of the users. You can always post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
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 $\Delta ABC$?
"Questions we can't answer are far better than answers we can't question"
Re: Dhaka 2014,Secondary,P6
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 $K=sr$.
Last edited by rah4927 on Wed Dec 09, 2015 2:44 pm, edited 1 time in total.
Re: Dhaka 2014,Secondary,P6
I think you wanted to tell that $D$ is the centre of the circle with centre on $BC$.rah4927 wrote:Let $D$ be the centre of the circle with centre on $AB$.
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.
"Questions we can't answer are far better than answers we can't question"
Re: Dhaka 2014,Secondary,P6
Yes, edited, thanks.tanmoy wrote:I think you wanted to tell that $D$ is the centre of the circle with centre on $BC$.rah4927 wrote:Let $D$ be the centre of the circle with centre on $AB$.