# If $S, S_1, S_2$ be the circles of radii 5,3 and 2 respectively. If $S_1$ and $S_2$ touch externally and they touch internally with $S$. [closed]

If $$S, S_1, S_2$$ be the circles of radii 5,3 and 2 respectively. If $$S_1$$ and $$S_2$$ touch externally and they touch internally with $$S$$. The radius of circle $$S_3$$ which touches externally with $$S_1$$ and $$S_2$$ and internally with $$S$$ is?

I tried making a diagram and figuring out, but cannot bring a relation.

## closed as off-topic by Saad, Paul Frost, Eevee Trainer, KReiser, Lord_FarinDec 27 '18 at 9:05

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## 1 Answer

Let $$C_1, C_2$$ be the centers of the circles $$S_1, S_2$$ and $$C$$ the center of the circle $$S$$. If the circle with center $$O$$ and radius $$r$$ touches the circles $$S_1, S_2$$ externally and $$S$$ internally, then we have $$C_1C_2 = 5$$, $$OC_1 = r+2$$, $$OC_2=r+3$$, $$OC=5-r$$. If in the triangle $$ABC$$, the point $$D$$ is on $$BC$$ such that $$BD:DC = m:n$$, then it is not difficult to show that $$(m+n)^2 AD^2 = (m+n)(m AC^2 + n AB^2) - mn BC^2$$ (See, Loney, Plane Trigonometry, Page 187, Ex 29).

Applying this to the triangle $$CC_1C_2$$, we have $$25(5-r)^2 = 5(2(r+2)^2 + 3(r+3)^2) - 150$$ from which we get $$r = 30/19$$.