AB is a line segment and M is its mid point. Semicircle are drawn with AM,MB and AB as diameters on the same side of line AB. A circle (O, r) is drawn so that it touches all 3 semi circles . prove that r=1/6 AB.

Answer 1

Let the length AM=1.
Let C be the midpoint of AM.
Draw the line CO. It has radius 1/2 + r.
Draw the line MO. It has length 1-r.

So, in the right triangle CMO,

1/2^2 + (1-r)^2 = (1/2 + r)^2