Question:
Segment $s_1$ has endpoints at $(3+\sqrt{2},5)$ and $(4,7)$.  Segment $s_2$ has endpoints at $(6-\sqrt{2},3)$ and $(3,5)$.  Find the midpoint of the segment with endpoints at the midpoints of $s_1$ and $s_2$.  Express your answer as $(a,b)$.

Answer:
Using the midpoint formula, we find that the midpoint of $s_1$ has coordinates $\left(\frac{3+\sqrt{2}+4}{2},\frac{5+7}{2}\right)=\left(\frac{7+\sqrt{2}}{2}, 6\right)$.

The midpoint of $s_2$ has coordinates $\left(\frac{6-\sqrt{2}+3}{2},\frac{3+5}{2}\right)=\left(\frac{9-\sqrt{2}}{2}, 4\right)$.

Applying the formula once more we see that the desired point is at $\left(\dfrac{\dfrac{7+\sqrt{2}+9-\sqrt{2}}{2}}{2},\frac{4+6}{2}\right)=\boxed{(4,5)}.$