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?
This machine computes the best location to view a drive in movie (maximizes the angle).
The machine is composed of a pair of mechanical jaws attached at the ends to the top and bottom of the screen and at the middle to the viewing plane. The jaws are pushed apart by a compressed spring, which will find equilibrium when the jaws are as far open as possible..
Move the points then press the play button. The potential energy of the (zero natural length) springs is proportional to the sum of their length.
Equilibrium will occur when the regression line is found
Drag A,B, C to change the original triangle. Press Play to watch a spring driven equilateral triangle gradually settle at the maximum perimeter position.
Press Show to see the perpendiculars to the sides of the equilateral triangle through A, B and C.
Note that the mechanism settles at a position where these perpendiculars are concurrent.
This is a problem from The Mathematical Gazette March 2014 pp 79-84.