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