= Rigidity of a scalene triangle
The vertex <set> of a triangle with three distinct side lengths has trivial setwise <stabilizer subgroup> in the plane <isometry group>. Each vertex is characterized by the unordered pair of lengths of its incident edges, so a setwise-preserving <Euclidean isometry> fixes every vertex. An <isometry> fixing three noncollinear plane points is the identity: squared <Euclidean distances> to them determine every other point, since subtracting two of the <Euclidean distance> equations gives two independent linear equations. For example, $(0,0),(1,0),(0,2)$ have side lengths $1,2,\sqrt5$.
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