Suppose you have an isosceles triangle with equal sides ST and TU of length n and a base SU of variable length m.
What is the "special" shape of the triangle of maximal area?
Source: Avital & Barbeau, "Intutively Misconceived Solutions to Problems," For the Learning of Mathematics, 1991
Hint: It is not equilateral....!
Use GSP to draw an isosceles triangle whose apex is the center of the circle and two radii form the equal sides. As you move a triangle vertex anchored on the circle, what happens to the area?
Solution Commentary: To overcome the first intuition of the answer being an equilateral triangle, the second intuition should be connected to other problems where you have maximal areas in polygons. Hint: Think square....and now can you prove your results?