Calculate the minimum of $\sqrt{(x-1)^{2}+y^{2}}+\sqrt{(x+1)^{2}+y^{2}}+|2-y|$ for $x, y \in \mathbb{R}$.
Let $A=(-1,0), B=(1,0)$ and $D$ be the line $y=2$. We need to minimize $M A+M B+d(M, D)$. The point $M$ necessarily belongs to the y-axis. We then reduce the problem to minimizing $2 \sqrt{1+y^{2}}+|2-y|$. We quickly see that $0 \leqslant y \leqslant 2$. A differentiation allows us to show that the minimum is achieved at $y=1 / \sqrt{3}$, and that this minimum is $2+\sqrt{3}$.