[ Extreme properties. Problems on maximum and minimum. Translation helps solve the task

Where should a bridge $MN$ be built across a river that separates two given villages $A$ and $B$, so that the path $AMNB$ from village $A$ to village $B$ is the shortest (the river banks are considered parallel lines, the bridge is assumed to be perpendicular to the river).

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Consider the images of points $A$ and $M$ under a parallel translation by the vector $\overrightarrow{M N}$ and use the triangle inequality.

## Solution

Suppose some position of the bridge is found. Under a parallel translation by the vector $\overrightarrow{M N}$, point $A$ will move to some point $A_{1}$, and point $M$ will move to point $N$. Then

$$
A M + M N + N B = A A_{1} + A_{1} N + N B \geqslant A A_{1} + A_{1} B
$$

(triangle inequality), and equality is achieved if points $A_{1}, N$, and $B$ lie on the same line, i.e., $B N \parallel A M$.

From this, the following construction method follows. From point $A$, lay off the segment $A A_{1}$, equal in length to the width of the river and perpendicular to its direction, and connect point $A_{1}$ with point $B$. The point $N$, obtained at the intersection of $A_{1} B$ with the bank of the river closer to $B$, will determine the position of the bridge.

![](https://cdn.mathpix.com/cropped/2024_05_06_1b164f828743b46f847cg-42.jpg?height=566&width=675&top_left_y=174&top_left_x=700)

## [ Parallel Translation. Constructions and Geometric Loci ]  Problem 55695 Topics: Diameter, Chords, and Secants Translation helps solve the problem.

Using a compass and a straightedge, draw a line parallel to a given line such that two given circles cut off equal-length chords on it.

## Hint

Apply a parallel translation.

## Solution

Suppose the required line is drawn. Let $A B = C D$ be the chords of the given circles $S_{1}$ and $S_{2}$, parallel to the given line $l$, and let $Q_{1}$ and $Q_{2}$ be the projections of the centers $O_{1}$ and $O_{2}$ of these circles onto the line $l$. Then, under a parallel translation by the vector $\overrightarrow{O_{1} Q_{2}}$, the segment $A B$ will move to the segment $C D$, and the circle $S_{1}$ will move to the circle $S$, which has a common chord $C D$ with the circle $S_{2}$.

From this, the following construction method follows. Drop perpendiculars from the centers of the given circles $S_{1}$ and $S_{2}$ to the given line $l$. Let $Q_{1}$ and $Q_{2}$ be the feet of these perpendiculars. If, under a parallel translation by the vector $\overrightarrow{Q_{1} Q_{2}}$, the image $S$ of the circle $S_{1}$ intersects the circle $S_{2}$ at two points $C$ and $D$, then $C D$ is the required line.

![](https://cdn.mathpix.com/cropped/2024_05_06_1b164f828743b46f847cg-42.jpg?height=494&width=834&top_left_y=1897&top_left_x=612)

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