In acute angled $\triangle ABC$, considering a portion of side $BC$ as diameter, a circle is drawn whose radius is 18 units and it touches two sides $AB$ and $AC$ respectively. 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 $\triangle ABC$?     Consider the diagram above. Here, $FL = 18, EI = 6$ and $DK = 9$ and yellow marked circle is the incircle.  I couldn't find a way out to determine each side of the triangle $ABC$. I researched with the diagram in Geogebra for a while, connecting or drawing some important segments but nothing seemed to me so much effective or useful.  Is there any formula by which I can get directly the radius of incircle with the given condition? Thanks in advance.  SOURCE: Bangladesh Math Olympiad
NOTE: I would appreciate it if you provided a hint and not the whole solution.  BdMO 2014 Nationals:     In acute angled triangle 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 ∆ABC?   The main problem with this  problem is that the figure is incredibly messy.So I drew one circle,and tried getting information from it.I noticed that if $O$ is a centre of one of the above (semi)circles,and if $O$ is located on $AB$, then $CO$ is an angle bisector.Therefore,if we connect all the vertices with the centres,their intersection would be the incentre.Then we need to drop a perpendicular from this point to get the inradius.But that hardly helps us.I have found some similar triangles in the figure.But that does not help me at all.  I have also tried backtracking.I tried to put myself in the problem-maker's shoes and tried to imagine what I would do if I had to create a problem like this.Unfortunately,I couldn't do it..  Any insightful comment,hint will be very much  appreciated.
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