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, thenThe 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 $(16/5,\pm 12/5)$. The two signs give reflected congruent triangles. This derivation explains both why three sides determine the shape and why mirror images remain possible.