By allowing the pendulum to wrap around a cycloid curve, we can constrain its mass to follow another cycloid (this, technically is because the evolute of a cycloid is a cycloid). As the cycloid is a tautochrone, the pendulums will have the same period no matter what their amplitude.
Try dragging one mass close to horizontal, while leaving the other close to vertical. Do you observe that they have the same period?

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?

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.

This simulation finds the Fermat point as the equilibrium position of the knot in 3 pieces of string, and as the intersection of the perpendiculars to a spring loaded equilateral triangle.
Try dragging the corners of the yellow triangle.