Ruby, 87 82 77 bytes

->*a{a.map!{|a,b,c|(w=[a-b,b-c,a-c].map &:abs).map{|x|x/w.sum}.sort}.uniq!=a}

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Given the 2 triangles as vectors of 3 complex numbers, calculate length of the three sides as distance between points, then normalize and check if the result is the same.

Ruby, 87 82 77 74 bytes

->*a{a.map!{|a,b,c|x,y,z=[a-b,b-c,a-c].map(&:abs).sort;[x/z,y/z]}.uniq!=a}

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Given the 2 triangles as vectors of 3 complex numbers, calculate length of the three sides as distance between points, sort ascending, then check if a/b and a/c are the same for both.

Ruby, 87 82 bytes

->*a{a.map{|a,b,c|(w=[a-b,b-c,a-c].map(&:abs)).map{|x|x/w.sum}.sort}.reduce(&:==)}

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Given the 2 triangles as vectors of 3 complex numbers, calculate length of the three sides as distance between points, then normalize and check if the result is the same.

Ruby, 87 82 77 bytes

->*a{a.map!{|a,b,c|(w=[a-b,b-c,a-c].map &:abs).map{|x|x/w.sum}.sort}.uniq!=a}

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Given the 2 triangles as vectors of 3 complex numbers, calculate length of the three sides as distance between points, then normalize and check if the result is the same.

Ruby, 87 bytes

->*a{!(a.map{|a,b,c|[a-b,b-c,a-c].map(&:abs).sort}.transpose.map{|a,b|a/b.to_f}|[])[1]}

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Ruby, 87 82 bytes

->*a{a.map{|a,b,c|(w=[a-b,b-c,a-c].map(&:abs)).map{|x|x/w.sum}.sort}.reduce(&:==)}

Try it online!

Given the 2 triangles as vectors of 3 complex numbers, calculate length of the three sides as distance between points, then normalize and check if the result is the same.