6. a) The radii of two circles \( S_1 \) and \( S_2 \) are \( r_1 \) and \( r_2 \), and the distance between their centers \( O_1 \) and \( O_2 \) is \( d \). What is the minimum distance between the points of the circles? What is the maximum distance between their points?

b) Answer the same question with the circles \( S_1 \) and \( S \) replaced by the disks \( K_1 \) and \( K_2 \) bounded by these circles.

It is clear that neither the maximum distance between the points of two figures nor the minimum of these distances characterizes the proximity of the figures to each other: for example, in Fig. 1a, the figures \( F_1 \) and \( F_2 \) are far from each other, but the minimum distance \( AB \) between their points is small; in Fig. 1b, the lines \( L_1 \) and \( L_2 \) almost coincide, but the maximum distance \( MN \) between their points is large. A much better characterization of the mutual position of two figures \( \Phi_1 \) and \( \Phi_2 \) is the following quantity. For each point \( A \) of the figure \( \Phi_1 \), find the closest point \( B \) of the figure \( \Phi_2 \) (Fig. 2); then determine the maximum distance \( A_0 B_0 \) among all such distances \( AB \). This distance \( A_0 B_0 \) is called the distance from the figure \( \Phi_1 \) to the figure \( \Phi_2 \) and is denoted by \( \rho(\Phi_1, \Phi_2) \).

Quantities whose construction is similar to the construction of the quantity \( \rho(\Phi_1, \Phi_2) \) are quite common in modern (especially applied) mathematics; therefore, it is appropriate to dwell in more detail on the process of forming the number \( \rho(\Phi_1, \Phi_2) \). The maximum of a set of numbers \( M = \{a, b, c, \ldots\} \) (the numbers of the set are listed in curly braces) is often denoted by the symbol

\[
\max [a, b, c, \ldots] \quad \text{or} \quad \max_{x \in M} x
\]

where the letters max are an abbreviation of the Latin word maximum - "greatest", and the notation \( x \in M \) is read as "the number \( x \) belongs to the set \( M \)", or "x is some number from the set \( M \)" ( \( \in \) is a set-theoretic symbol of membership of an object in a given collection, or set of elements). Similarly, the minimum of the same set of numbers \( M = \{a, b, c, \ldots\} \) is denoted as follows:

\[
\min [a, b, c, \ldots] \quad \text{or} \quad \min_{x \in M} x
\]

where min is an abbreviation of the Latin word minimum - "smallest". We have seen above that the quantities

\[
\min_{A_1 \in \Phi_1, A_2 \in \Phi_2} A_1 A_2 \quad \text{and} \quad \max_{A_1 \in \Phi_1, A_2 \in \Phi_2} A_1 A_2
\]

(Fig. 1)

(Fig. 2)

(the first notation is read as "the minimum of the distances \( A_1 A_2 \), where the point \( A_1 \) belongs to the figure \( \Phi_1 \), and the point \( A_2 \) belongs to the figure \( \Phi_2 \); the second notation has an analogous meaning) are poor characteristics of the deviation of the figure \( \Phi_2 \) from the figure \( \Phi_1 \). A much better fit for this purpose is the quantity \( \rho(\Phi_1, \Phi_2) \), which is defined as follows: first, fixing some point \( A \) (which will subsequently be considered to belong to the figure \( \Phi_1 \)), we form

\[
\min_{B \in \Phi_2} A B
\]

i.e., the minimum of all distances \( A B \), where \( B \) is any point of the figure \( \Phi_2 \); the resulting quantity is denoted by \( \rho(A, \Phi_2) \) and is often called the distance from the point \( A \) to the figure \( \Phi_2 \). Then we find the quantity

\[
\max_{A \in \Phi_1, B \in \Phi_2} A B \quad \text{(or more precisely, } \max_{A \in \Phi_1} \left( \min_{B \in \Phi_2} A B \right) \text{),}
\]

i.e., the maximum of all possible quantities \( \min_{B \in \Phi_2} A B \), assuming that \( A \) runs through all points of the figure \( \Phi_1 \); this "maximum of all minima" is denoted by \( \rho(\Phi_1, \Phi_2) \).

In the future, in this book, we will encounter quantities formed in a similar manner several times (see, for example, p. 24 or 35).

Finally, let us stop at one more point, although it is not essential for the problems of this cycle, but is worth attention from the point of view of the problems discussed in subsequent cycles. By a figure, we mean any set of points; but in this case, for example, already \( \min A B \), where \( A \) is some fixed point and \( B \in \Phi \) is a (planar) figure, may not exist. Indeed, let \( K \) be, for example, the set of all interior points of a circle with center \( O \) and radius \( r \) (i.e., such points \( M \) that \( O M < r \)); then for any point \( A \) outside the circle, the minimum distance \( \min A B \) does not exist, because there are points \( M \in K \) such that \( A M \) can be arbitrarily close to \( A N_0 \), but among the points \( M \in K \) there will be points for which \( A M \) is arbitrarily close to \( A N_0 \). In the case when

1) Recall, for example, the well-known definition of the distance from a point \( A \) to a line \( a \) as the length \( d \) of the perpendicular dropped from \( A \) to \( a \); it is clear that \( d = \rho(A, a) \),
\( \min A B \) does not exist, the distance \( \rho(A, \Phi) \) from \( A \) to \( \Phi \) is defined as (the) lower bound, or infimum, of the lengths of all segments \( A B \), where \( B \in \Phi \):

\[
\rho(A, \Phi) = \inf_{B \in \Phi} A B
\]

where the quantity \( \xi = \inf_{x \in X} x \), where \( X \) is some set of numbers, is defined by the conditions: \( x \geq \zeta \) for all \( x \in X \); there exist \( x \in X \) arbitrarily close to \( \zeta \) (so if \( \min x \) exists, then \( \inf_{x \in X} x = \min_{x \in X} x \)). Similarly, in cases where \( \max_{A \in \Phi_1} \rho(A, \Phi_2) \) does not exist, it is often convenient to consider (the) upper bound, or supremum, of the quantities \( \rho(A, \Phi_2) \), where \( A \in \Phi_1 \):

\[
\rho(\Phi_1, \Phi_2) = \sup_{A \in \Phi_1} \rho(A, \Phi_2) = \sup_{A \in \Phi_1} \inf_{B \in \Phi_2} A B
\]

where \( \sup y \) is defined as the number \( \eta \) such that \( y \leq \eta \) for all \( y \in Y \) but the difference \( \eta - y \), where \( y \in Y \), can be arbitrarily small. If the distance \( \rho(\Phi_1, \Phi_2) \) from the figure \( \Phi_1 \) to the figure \( \Phi_2 \) is defined as:

\[
\rho(\Phi_1, \Phi_2) = \sup_{A \in \Phi_1} \inf_{B \in \Phi_2} A B
\]

then, for example, the distance between the interiors of two circles will not differ from the distance between the circles themselves.
6. a) Let the points of intersection of the circles $S_{1}$ and $S_{2}$ with their line of centers $O_{1} O_{2}$ be denoted by $A_{1}, B_{1}$ and $A_{2}, B_{2}$, with the point $O_{2}$ lying on the ray $O_{1} B_{1}$, and the point $O_{1}$ on the ray $O_{2} B_{2}$ (Fig. $63, a-c$; if the circles $S_{1}$ and $S_{2}$ are concentric, then any line passing through the common center can be chosen as the line of centers

![](https://cdn.mathpix.com/cropped/2024_05_21_d4ffc236376258137647g-126.jpg?height=231&width=438&top_left_y=304&top_left_x=150)
a)

![](https://cdn.mathpix.com/cropped/2024_05_21_d4ffc236376258137647g-126.jpg?height=293&width=384&top_left_y=538&top_left_x=205)
b)

![](https://cdn.mathpix.com/cropped/2024_05_21_d4ffc236376258137647g-126.jpg?height=329&width=352&top_left_y=302&top_left_x=674)

b)

Fig. 63.

$S_{1}$ and $S_{2}$ of a straight line and denote by $A_{1}$ any point of its intersection with the circle $S_{1}$, and by $A_{2}$ any point of its intersection with the circle $S_{2}$). Further, let $M_{1}$ and $M_{2}$ be arbitrary points on the circles $S_{1}$ and $S_{2}$. From the consideration of the quadrilateral $O_{1} O_{2} M_{2} M_{1}$, it is clear that

$M_{1} M_{2} \leqslant M_{1} O_{1} + O_{1} O_{2} + O_{2} M_{2}$, or $\quad M_{1} M_{2} \leqslant A_{1} O_{1} + O_{1} O_{2} + O_{2} A_{2}$, i.e.,

$$
M_{1} M_{2} \leqslant A_{1} A_{2}
$$

from which it follows that the greatest distance between the circles is equal to the segment $A_{1} A_{2}$, i.e., $d + r_{1} + r_{2}$. On the other hand, if the circles $S_{1}$ and $S_{2}$ do not intersect, i.e., $d > r_{1} + r_{2}$ (Fig. $63, a$), or $d < r_{1} - r_{2}$ (Fig. $63, \bar{\sigma}$; here we assume for definiteness that $r_{1} \geqslant r_{2}$), then from the same quadrilateral $O_{1} O_{2} M_{2} M_{1}$ it follows that

$$
O_{1} O_{2} \leqslant O_{1} M_{1} + M_{1} M_{2} + M_{2} O_{2}
$$

or

$$
M_{1} M_{2} \geqslant O_{1} O_{2} - O_{1} M_{1} - O_{2} M_{2} = O_{1} O_{2} - O_{1} B_{1} - O_{2} B_{2},
$$

i.e., $M_{1} M_{2} \geqslant B_{1} B_{2}$ (Fig. $63, a$), respectively, that

$$
O_{1} M_{1} \leqslant O_{1} O_{2} + O_{2} M_{2} + M_{2} M_{1}
$$

or

$$
M_{1} M_{2} \geqslant O_{1} M_{1} - O_{1} O_{2} - O_{2} M_{2} = O_{1} B_{1} - O_{1} O_{2} - O_{2} A_{2}
$$

i.e., $M_{1} M_{2} \geqslant B_{1} A_{2}$ (Fig. 63,6). Thus, in the case of Fig. 63, $a$ the smallest distance is realized by the segment $B_{1} B_{2}$, and in the case of Fig. $63, \sigma$ - by the segment $B_{1} A_{2}$, so that this smallest distance is equal to $d - r_{1} - r_{2}$ (Fig. $63, a$) or $r_{1} - d - r_{2}$ (Fig. 63, b). Finally, if the circles $S_{1}$ and $S_{2}$ intersect (Fig. $63, \boldsymbol{s}$) or touch, then the smallest distance between the points of the circles is zero.

b) It is clear that the greatest distance between the points of the circles $K_{1}$ and $K_{2}$ in all cases is equal to the greatest distance between the points of the circles $S_{1}$ and $S_{2}$ bounding these circles. The smallest distance between the points of the circles differs from zero only in the case when the circles bounding the circles lie outside each other (see Fig. $63, a$), and in this case it is equal to the smallest distance between the points of the circles.