Consider a series of n concentric circles $c_1,c_2 \cdots c_n$ with radii $r_1,r_2.\cdots r_n$ 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 $c_{i+2} (i =1,2, \cdots)$ Find the value of $\lim_{n \to \infty} \sum^n_{i=1} r_i$ if the angle between tangents from any point of $c_1$ to $c_2$ is $\frac{\pi}{3}$
Please suggest how to proceed in this case thanks.
Let us trace the two tangents from a point C of c1 to the smaller circumference c2, and let us call A and B the contact points. We can also trace two other lines from A and B to the center O, and a third line from the initial point C to O.
Now we have two identical right triangles with a common hypothenuse, in which the angles ACO and BCO are $\pi/6$ and the angles AOC and BOC are $\pi/3$. Therefore, the length of the hypothenuse CO (radius of c1) is equal to twice the length of the short legs AO and BO (radii of c2). Moreover, calling D the point where the chord AB intersect the hypothenuse CO, it is not difficult to show that DO (radius of c3), being the projection of the short legs AO and BO on the hypothenuse, is equal to half the radius of c2.
In summary, calling $r_1$, $r_2$, and $r_3$ the radii of $c_1$, $c_2$, and $c_3$, respectively, we have $c_3=1/2\,r_2= 1/4\,r_1$. Proceding in this way, we get that the limit of the sum of all radii is $2r_1$, i.e., 20.