AB is a line segment and M is its midpoint. Semicircles are drawn with AM, MB and AB as diameters on the same side of line AB. A circle is drawn to touch all the semicircles. Prove that its radius r is given by AB/6.

Let the length of AB=x, and radius of the inner circle be r.AM =MB=x/2AL=PL=QN=x/4OM ABHence from pythagoras theoremOL2=OM2+LM2OL =(r+x4)OM = (RM-OM)=(x2-r)LM=x4(r+x4)2=(x2-r)2+(x4)2r2+rx2+x216=x24-rx+r2+x216rx2=x24-rxr2=x4-r3r2=x4r=x6Hence Radius r =AB6 Ans

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