MathDB

(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)

2