The distance between the centers of circles with radii $r$ and $R$ is $a$, and $a>r+R$. Find the smallest distance between points, one of which lies on the first circle, and the other on the second (the distance between the circles

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Using the generalized triangle inequality, prove that the distance between any two points on these circles is not less than $a-r-R$.

## Solution

Let $O_{1}$ and $O_{2}$ be the centers of circles with radii $r$ and $R$ respectively, and the line connecting the centers intersects the circles at points $A$ and $B$, respectively, such that $A$ and $B$ lie between $O_{1}$ and $O_{2}$. Then, if $X$ and $Y$ are any points lying on the first and second circles, respectively, we have

$$
X O_{1} + X Y + Y O_{2} \leqslant O_{1} O_{2} = A O_{1} + A B + B O_{2}
$$

or

$$
r + X Y + R \leqslant r + A B + R
$$

Therefore, $X Y \leqslant A B = a - r - R$, and equality is achieved only when $X = A$ and $Y = B$.

![](https://cdn.mathpix.com/cropped/2024_05_06_cb0f0d605674d837f9abg-18.jpg?height=312&width=627&top_left_y=2099&top_left_x=724)

Answer

$a - r - R$.

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