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A circle C has radius 6. Two circles P and Q having equal radii are touching each other externally and touching C internally. Another circles S is touching the circle C internally and the circles P, Q externally. The radius of the circle S is
For S to be fixed, P and Q must each have radius 3, with centres $$(\pm 3, 0)$$ on a diameter of C (centre at the origin). Let S have radius $$s$$ and centre $$(0, 6 - s)$$. Touching P externally gives $$\sqrt{9 + (6 - s)^2} = 3 + s$$, so $$45 - 12s + s^2 = 9 + 6s + s^2$$, i.e. $$18s = 36$$ and $$s = 2$$.
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