Constant force actuators pull the masses and remain tangent to the curve. Potential energy for these actuators corresponds to their length. Hence a minimal energy configuration should minimize the perimeter of the triangle.
Press Show to see the cevians to the points of contact with the curve.
Pres Show a second time to see the normals at the points of contact.
What do you notice?

Constant force actuators push outwards. As the potential energy in each actuator is equal to its length, this machine should settle where the triangle's perimeter is maximum.
Press Show to see the reflections in the tangents of a beam of light transmitted along one edge of the triangle.

A force proportional to the length of each side are applied to the side's midpoint. This models a constant pressure, and potential energy should be minimized when area is maximized.
Press Show to see the tangents to the curve at the points of contact.
What do you notice?
You can change the definition of the curves, but be careful, or the masses might escape!