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Point of Equal Tension

Drag points A, B, and C and observe what happens to point P.

Notice that as points A, B, and C are dragged around, angles of 120° are preserved where the segments intersect at point P. Another way to understand point P is by imaging the three segments to be three lengths of rope connected at a very flexible knot at point P. If people at points A, B, and C all pull on the rope with equal tension, then the knot at point P will stabilize when all the angles are 120°. A third way to understand point P is that the sum AP + BP + CP of the distances from P to the other points is minimized when P is at the 120°-point of the triangle ∆ABC. You can look into my construction of this 120°-point of a triangle by playing with the custom tools I created (wrench icon, far right of the toolbar). Note that due to my own limitations with Geogebra, there are some issues with the construction of point P when you try to create a triangle with an angle greater than 120°.