# Can use Coordinate geometry

Geometry Level 4

Consider a series of $$n$$ concentric circles $${C}_{1}, {C}_{2}, \ldots, {C}_{n}$$ with radii $${r}_{1}, {r}_{2}, {r}_{3}, \ldots , {r}_{n}$$ respectively satisfying $${r}_{1}>{r}_{2}>{r}_{3}> \cdots >{r}_{n}$$ and
$${r}_{1} = 10$$.

The circles are such that the chord of contact of tangents from any point on $${C}_{i}$$ to $${C}_{i+1}$$ is a tangent to $${C}_{i+2}$$ where $$i = 1, 2, 3, ...$$.

Find the value of $$\displaystyle \ \lim_{ n\to \infty }{ \sum _{ r=1 }^{ n }{ { r }_{ i } } }$$, if the angle between the tangents from any point of $${C}_{1}$$ to $${C}_{2}$$ is $$60^\circ$$.

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