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.