Using Exercise 14, show algebraically that if two plane isometries and agree on three noncolinear

Using Exercise 14, show algebraically that if two plane isometries ∅ and ψ agree on three noncolinear points, that is, if ∅(P_i) = ψ(P_i) for noncolinear points P₁, P₂ , and P₃ , then∅ and ψ are the same map.

Data from exercise 14.

Argue geometrically that a plane isometry that leaves three noncolinear points fixed must be the identity map.