I have the following figure
Where $AB=10$m, $BD=12$m and $DE=12$m. The point C can slide along the segment BD. Now the problem is to minimize the distance from A to D going along the dashed line. The problem can be solved using simple analysis and differentiation. Let $BC=x$ then $\|ACE\| = f(x) = \sqrt{10^2-x^2}+\sqrt{(12-x)^2+12^2}$. in order to find a minima which we know exists one would have to solve $f'(x)=0$ to obtain the solution $x=60/11$.
My question is however can one prove without analysis that x=60/11 of $BD$ in order to minimize the distance $\|ACE\|$?