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Replacing Intuitive Notions With Second-Thought Intutions

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?