The Fermat Toricelli Point of a triangle is the point which minimizes the sum of the distances to the vertices of the triangle.
This app represents a mechanical device for computing the point.
Balls dangling from strings which pass through the vertices of the triangle naturally settle at the point of lowest potential energy. Can you see why this solves the Fermat Toricelli Problem?
We use rotations to construct a path equal in length to the sum of the distances AE+BE+CE, whose end points are fixed. Minimum for AE+BE+CE occurs when this path is a straight line.
So long as no angles of the original triangle are greater than 120 degrees, this works. What goes wrong when an angle does exceed 120 degrees?