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Let C be a plane convex body. For arbitrary points ,* a,b ∈E*^{ n}denote by │*ab*│ the Euclidean length of the line-segment ab. Let *a*_{1}*b*_{1} be a longest chord of C parallel to the line-segment ab. The relative distance *d*_{c}*(a,b)* between the points *a* and *b *is the ratio of the Euclidean distance between a and b to the half of the Euclidean distance between *a*_{1} and *b*_{1}. In this note we prove the triangle inequality in *E*^{2} with the relative metric *d*_{c( }^{.}_{,}^{.}_{)}, and apply this inequality to show that *6≤l(P)≤8*, where *l(P)* is the perimeter of the convex polygon *P *measured in the metric *d*_{p}_{( }^{.}_{,}^{.}_{)}. In addition, we prove that every convex hexagon has two pairs of consecutive vertices with relative distances at least 1.