Important Note: The question is at the very bottom.
The following link here states the following definition of an isometry in analytic geometry.
A more formal definition of congruence states that two subsets $A$ and $B$ of Euclidean space $R^n$ are called congruent if there exists an isometry $f : \mathbb{R}^n → \mathbb{R}^n$ with $f(A) = B$. [I want to use this definition for congruence of shapes other than triangles.]
Say we have to triangles $\triangle ABC$ and $\triangle A'B'C'$ that satisfy SAS where angles $A$ and $A'$ have the same measure for the angle part. I want to show this satisfies the definition of congruent figures above.
The figure for $\triangle ABC$ would be (using vectors): $\{\overrightarrow{AB}t+A| t\in \mathbb{R} \text{ and } 0 \le t\le 1\}\cup \{\overrightarrow{BC}t+B| t\in \mathbb{R} \text{ and } 0 \le t\le 1\}\cup \{\overrightarrow{CA}t+C| t\in \mathbb{R} \text{ and } 0 \le t\le 1\}$.
Also, the figure for $\triangle A'B'C'$ would be: $\{\overrightarrow{A'B'}t+A'| t\in \mathbb{R} \text{ and } 0 \le t\le 1\}\cup \{\overrightarrow{B'C'}t+B'| t\in \mathbb{R} \text{ and } 0 \le t\le 1\}\cup \{\overrightarrow{C'A'}t+C'| t\in \mathbb{R} \text{ and } 0 \le t\le 1\}$.
We translate point $A$ in $ABC$ to $A'$. We do the same procedure with all other points in the figure $ABC$. Translating our shape from $ABC$ this way is an isometry. We now rotate our triangle so that $AC$ lies directly on $A'C'$. Rotation is again an isometry. We then reflect our triangle if needed so that triangle $ABC$ lies directly on top of $A'B'C'$. Reflection is an isometry. The set of points in both figures are now equal throughout the transformation. As each transformation is bijective, we can make a compositional map of distance preserving functions from triangle $ABC$ to $A'B'C'$.
$\textbf{Question:}$ How is this process in analytic geometry (using transformations) not considered to be superposition? I always thought superposition was considered to be a bad thing and should never be done. However, now I am thinking it is preferred for showing congruence of harder shapes such as polygons.