The Steiner point S of a triangle ABC is such that the total distance SA + SB + SC is as small as possible. The nice fact about S is that the angles ASB, BSC and CSA are all 120°. I mentioned this casually in a lesson with 5Ma recently, and they decided we would try to prove it. It became clear quite quickly that we would have to invent a bit of partial differentiation, but nothing too advanced.