If A = (0,0), B = (6,0) and C = (3,4) then AB>AC in Euclidean distance (because 6>5) but AC>AB in Manhattan distance (because 3+4=7>6).

If A = (0,0), B = (6,8) and C = (9,0) then AB>AC in Euclidean distance (because 10>9) but AC>AB in Chebyshev distance (because max{6,8}=8<9).