NO-ABEL 2012 Problem 2

(a)Two circles $S_1$ and $S_2$ are placed so that they do not overlap each other, neither completely nor partially. They have centres in $O_1$ and $O_2$ , respectively. Further, $L_1$ and $M_1$ are different points on $S_1$ so that $O_2L_1$ and $O_2M_1$ are tangent to $S_1$ , and similarly $L_2$ and $M_2$ are different points on $S_2$ so that $O_1L_2$ and $O_1M_2$ are tangent to $S_2$ . Show that there exists a unique circle which is tangent to the four line segments $O_2L_1, O_2M_1, O_1L_2$ , and $O_1M_2$ .

(b) Four circles $S_1, S_2, S_3$ and $S_4$ are placed so that none of them overlap each other, neither completely nor partially. They have centres in $O_1, O_2, O_3$ , and $O_4$ , respectively. For each pair $(S_i, S_j )$ of circles, with $1 \le i < j \le 4$ , we find a circle $S_{ij}$ as in part a. The circle $S_{ij}$ has radius $R_{ij}$ . Show that $\frac{1}{R_{12}} + \frac{1}{R_{23}}+\frac{1}{R_{34}}+\frac{1}{R_{14}}= 2 \left(\frac{1}{R_{13}} +\frac{1}{R_{24}}\right)$