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The point which minimizes weighted sum of distances to triangle's vertices

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Phil Todd

@polymnia

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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?

Tags: fermat, toricelli, triangle
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