Distance measure in a weighted, fully connected network

Hey,

I have a network where the adjacency matrix (A) is a transformation of the correlation coefficients and self-edges are removed

A=0.5*(correlation + 1)

Now I need a distance measure to calculate betweenness and thought about transforming it to a distance d by

d=(1-A)^2

How can I check whether the triangle inequality holds for this distance? All other conditions of a metric are satisfied, but I'm unsure how to prove this. I need the distance to calculate betweenness scores for the network and the applied distance measure spreads the distribution of distance so that I get results other than 0.

However, my problem of proving whether the triangle inequality holds remains.

Thank you, Chris