The Fundamental Theorem of Similarity Geometry If L = (A,B) and L' = (A',B') are lists of distinct points of E, then there is a unique element of S^{+}(2) which maps L to L'.


Proof Let A, B, A', B' have complex coordinates α, β, α', β' respectively. An element s of S^{+}(2) has the form s(z) = γz+δ, with γ ≠ 0. This maps L to L' if and only if γα + β = α', and γβ + δ = β'. As A, B are distinct α ≠ β, and as A', B' are distinct, α' ≠ β'. As (α  β) ≠ 0, the equations have the unique solution γ = (α'  β')/(α  β) and δ = (αβ'  α'β)/(αβ). As α' ≠ β', γ ≠ 0, so s(z) = γz + δ is in S^{+}(2). Thus we have a unique element of S^{+}(2) mapping L to L'.
This is a purely algebraic proof, and gives no idea how the map can be constructed.
We begin with a dilation d about A mapping B to C, where AC = A'B'. Unfortunately, the uniqueness of s is not at all clear! 