[TomCowley]

Given a triangle ABC, area = 1/2AB^2(Sin A)(Sin B)/(Sin C). For any triangle, holding AB and Sin C (the opposite angle) constant, and performing an infinitesimal change in angles A and B has the following property:

Because the second derivative of Sin is negative throughout the legal range of angles, bringing the angles infinitesimally closer together while keeping their sum the same increases the product (area), moving them further apart decreases the product (area).

So the maximum area is when the angles are equal, so any non-equilateral triangle does not maximize area for a given perimeter, which is equivalent to stating that perimeter is not minimized for a given area.