Background: I am trying to prove that reflections are an isometry, i.e., $d(P,Q)=d(f(P),f(Q))$ when $f$ is a reflection across the line $\ell$.
Main question: Is there a proof to state that the lines $\overline{PQ}$ and $\overline{f(P)f(Q)}$ both intersect $\ell$ at the same point $C$? I have included an illustration below with these labels.
Additional info: My main goal is to prove that the distances of $[P,Q]$ and $[f(P),f(Q)]$. If I can prove the above, I will then use Thales' Theorem and the fact that $[C,P,Q]$ is an isosceles triangle to show that these distances are indeed equal.