Let $x,$ $y,$ and $z$ be positive real numbers.  Find the minimum value of
\[\frac{\sqrt{x^2 + y^2} + \sqrt{x^2 + z^2} + \sqrt{y^2 + z^2}}{x + y + z}.\]
Question:
Let $x,$ $y,$ and $z$ be positive real numbers.  Find the minimum value of
\[\frac{\sqrt{x^2 + y^2} + \sqrt{x^2 + z^2} + \sqrt{y^2 + z^2}}{x + y + z}.\]
Answer:
Let's think step by step.

user
\sqrt{2}
Let $x,$ $y,$ and $z$ be positive real numbers.  Find the minimum value of
\[\frac{\sqrt{x^2 + y^2} + \sqrt{x^2 + z^2} + \sqrt{y^2 + z^2}}{x + y + z}.\]
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