Let C be a plane convex body. For arbitrary points , a,b ∈E ndenote by │ab│ the Euclidean length of the line-segment ab. Let a1b1 be a longest chord of C parallel to the line-segment ab. The relative distance dc(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 a1 and b1. In this note we prove the triangle inequality in E2 with the relative metric dc( .,.), 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 dp( .,.). In addition, we prove that every convex hexagon has two pairs of consecutive vertices with relative distances at least 1.