That depends on your clustering algorithms. If it is for example based on shared-neighbor information, then you end up with the same clusters as you did before

That depends on your clustering algorithms. If it is for example based on shared-neighbor information, then you end up with the same clusters as you did before scaling. And in your example, I get: (A, B) < (B, C) < (A, C) before and (A, B) < (B, C) < (A, C) after scaling. So the order is preserved. It really depends on your clustering algorithm. DBSCAN for example could end up with a different result due to its epsilon-range parameter.