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Let $X$ be the set of lines in the Euclidean plane. For $r,s$ $\in$ $X$, $d (r, s)$ is the Euclidean usual distance between $r$ and $s$ if the two lines are parallel, otherwise the arc angle (with value in $[0,\pi/2]$) of the smallest angle formed by rays if the two lines are incident. Than $(X,d)$ is a metric space?

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up vote 1 down vote accepted

HINT: What if $r$ and $s$ are parallel lines one unit apart, and $t$ is a line that makes a very small angle with $r$ and $s$? Do you have the triangle inequality $d(r,s)\le d(r,t)+d(t,s)$?

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Thank you for your hint :-) So this isn' t a metric space because it doesn't always respect the triangle inequality (in particular if r and s parallel and d(r,s)>pi and t a line with little angle with r and s) – Agenog Jan 23 '13 at 2:33
@Giacomo: That’s right. The other requirements for a metric are all satisfied; it’s only the triangle inequality that can fail. – Brian M. Scott Jan 23 '13 at 2:35

The triangle inequality does not hold. Take two parallel lines r and s and a line with very little slope with respect to them...

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