section_path: Applications/Geometry/Shapes
section_name: Shapes
paragraph_text: Shapes
  
  Three non-collinear points u, v, w {\displaystyle u,v,w} in the plane determine the shape of the triangle { u, v, w } {\displaystyle \{u,v,w\}}. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as S (u, v, w) = u − w u − v. {\displaystyle S(u,v,w)={\frac {u-w}{u-v}}.} The shape S {\displaystyle S} of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an affine transformation), corresponding to the intuitive notion of shape, and describing similarity. Thus each triangle { u, v, w } {\displaystyle \{u,v,w\}} is in a similarity class of triangles with the same shape.