Are my triangles similar?
MATL, 15 12 bytes
,i4:)d|S]/da

The program inputs two  3×1 vectors of complex numbers representing the coordinates; and outputs 0 for similar, 1 for not similar.
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Explanation
The code checks if the side lengths, sorted for each triangle, are proportional between the two triangles.
,      % Do twice
  i    %   Take input: 3×1 vector of complex numbers
  4:)  %   Modular index to repeat 1st number after the 3rd. Gives a 4×1 vector
  d    %   Consecutive differences
  |    %   Absolute value, element-wise
  S    %   Sort
]      % End
/      % Divide, element-wise
d      % Consecutive differences
a      % Any: gives 0 if and only if all values are 0
       % Implicit display