Are my triangles similar? Given (in any structure; flat list, two lists of lists, a tuple of matrices, a 3D array, complex numbers,…) the coordinates for two non-degenerate triangles ABC=[[Ax,Ay],[Bx,By],[Cx,Cy]] and PQR=[[Px,Py],[Qx,Qy],[Rx,Ry]], determine if they are similar, that is,


  they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection.[Wikipedia]


You may assume that all coordinates are integers.

You must either return a truthy/falsey value indicating similar/dissimilar respectively, or two consistent values; please state your choice.

Failing on some cases due to limitations in floating point precision  is acceptable so long as the algorithm is correct in principle.

Bonus task: Add a comment to this post stating whether you would like the same challenge generalised to polygons in N-space.

Walked-through example case

ABC=[[0,0],[1,0],[0,1]] and PQR=[[1,0],[-1,0],[1,-2]]


Reflect ABC in the x-axis: [[0,0],[-1,0],[0,1]]
Reflect in the y-axis: [[0,0],[-1,0],[0,-1]]
Enlarge by a factor of 2: [[0,0],[-2,0],[0,-2]]
Translate right by 1 unit: [[1,0],[-1,0],[1,-2]]


This gives us PQR.

Test cases

Similar

[[8,4],[5,-5],[0,0]] and [[-4,-1],[5,-1],[-1,5]]

[[-2,1],[4,-2],[6,2]] and [[-1,-1],[2,-1],[-1,1]]

[[-1,0],[1,0],[0,2]] and [[-2,5],[2,5],[0,1]]

Dissimilar

[[0,0],[-1,0],[0,1]] and [[1,0],[-1,0],[2,-3]]

[[2,1],[4,2],[6,2]] and [[-1,-1],[2,-1],[-1,1]]

[[-1,0],[1,0],[0,3]] and [[-2,5],[2,5],[0,1]]
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.
Try it online! Or verify all test cases.
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