In the plane of a given triangle $A_1A_2A_3$ determine (with proof) a straight line $l$ such that the sum of the distances from $A_1, A_2$, and $A_3$ to $l$ is the least possible.
1. **Initial Setup**: Consider a triangle \( A_1A_2A_3 \) in the plane. We need to find a line \( l \) such that the sum of the perpendicular distances from \( A_1 \), \( A_2 \), and \( A_3 \) to \( l \) is minimized.

2. **Transposition Argument**: Suppose the line \( l \) divides the plane into two regions, \( a \) and \( b \), with at least two points of the triangle in region \( a \). If we transpose the line \( l \) towards region \( a \) until it passes through one of the points, the sum of the distances from the points to the line will decrease. This is because moving the line closer to the points in \( a \) reduces their perpendicular distances to the line.

3. **Line Passing Through a Vertex**: For the smallest possible sum of distances, the line \( l \) must pass through one of the vertices of the triangle, say \( A_1 \), and separate the other two points \( A_2 \) and \( A_3 \).

4. **Dependence on Angle**: The sum of the distances from \( A_2 \) and \( A_3 \) to the line \( l \) is dependent on the angle \( \alpha \) formed by the line and the side \( A_2A_3 \). Specifically, the sum of the distances is given by \( A_2A_3 \sin \alpha \).

5. **Minimizing the Angle**: The smallest possible value of \( \alpha \) occurs when the line \( l \) is perpendicular to \( A_2A_3 \). However, since the line must pass through one of the vertices, the line that minimizes the sum of the distances is the one that coincides with one of the sides of the triangle.

6. **Conclusion**: Therefore, the line \( l \) that minimizes the sum of the distances from \( A_1 \), \( A_2 \), and \( A_3 \) to \( l \) is one of the sides of the triangle. The sum of the distances in this case is equal to the height of the triangle corresponding to the chosen side.

The final answer is the line that coincides with one of the sides of the triangle \( \boxed{ A_1A_2A_3 } \).