Search Results for “Fermat”
Fermat Toricelli
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?
Two ways of finding the Fermat Point
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.
Mechanical Location of the Fermat Toricelli Point
The Fermat Toricelli Point minimizes the sum of the distances to the vertices of a triangle. This app simulates a physical method for finding the point. Press Hint to see a geometric method.
Fermat Toricelli Geometric Proof
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?
The point which minimizes weighted sum of distances to triangle's vertices
In the mechanical simulation, the actuator connecting point D to A exerts a constant force a, the actuator connecting D to B exerts force b, and the actuator connecting D to C exerts force c. At equilibrium, potential energy will be minimized. The PE is proportional to a|AD|+b|BD|+c|CD| Press the hint button to see a geometric construction for this point. Under some circumstances the mass settles at one of the vertices. Can you work out what these circumstances are?
Weighted sum of distances
The app displays the weighted sum of the distances between point D and the vertices of the triangle, where the weights are a, b, c.
Largest Outscribed Equilateral Triangle
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.
