Sunday, January 3, 2010

Fermat Point

Sticking with the name of my blog, the Fermat Point was the first point of concurrency to be spotlighted. Not sure why I chose this point...I probably should have followed Clark Kimberling's Encyclopedia of Triangle Centers and chosen the Incenter, which is center X(1) in the ETC. The Gergonne Point is the next to be spotlighted, and it is closely related to the Incenter, so I think that could count.

The Fermat point, also known as the Torricelli point, is the point with the property that the total distance from the three vertices of the triangle to the point is as small as possible. The point takes its name from Fermat, who proposed this problem to Torricelli. More on the Fermat Point can be found at Cut-the-Knot. I use this with my geometry students after we study inscribed angles.

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1 comment:

  1. Steve, Some of my favorite notes about the Fermat Point...The Fermat point is also the point of intersection of the circumcircles of the three equilateral triangles. If we label A' as the new vertex of the of the equilateral triangle created on BC and similarly for B' and C', then the lengths of the segments A-A', B-B' and C-C' are all equal, AND>>> they are all the sum of PA, PB, and PC.

    By the way, an interesting thing to try::::: take an equilateral triangle with all three vertices free to move on the circumcircle... then hold two points constant and move the third across the 240 degrees of its freedom between the other two..then return to its original point, and do the same with a second and then the third... all the while tracing the locus of the Fermat Point.