= Side-side-side congruence
Side-side-side congruence states that two nondegenerate <triangles> in <Euclidean geometry> are <congruent> when their three corresponding side <lengths> agree. Congruence permits a <reflection>; the conclusion does not require an orientation-preserving motion.
To see uniqueness, put one side of length $c>0$ at $(0,0)$ and $(c,0)$. If the third vertex $(x,y)$ has distances $b$ and $a$ to those endpoints, then
$$
x^2+y^2=b^2,\qquad (x-c)^2+y^2=a^2,\qquad x=\frac{b^2+c^2-a^2}{2c}.
$$
The value of $y^2$ is now fixed, leaving only the two reflected choices of sign. Strict <triangle inequalities> ensure $y^2>0$.
For side lengths $3,4,5$ with $c=5,b=4,a=3$, the third vertex is $(16/5,\pm12/5)$. The two signs give reflected congruent triangles. This derivation explains both why three sides determine the shape and why mirror images remain possible.
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