The Fermat Point

The Fermat point is named for the point which is the solution to a geometric challenge that Pierre Fermat posed for Evangelista Torricelli, who was briefly an associate of the aged Galileo. Fermat challenged Torricelli to find the point P in an acute triangle ABC which would minimize the sum of the distances to the vertices A, B, and C. The triangle need not actually be acute, but if the largest angle reaches 120 o or more, then the vertex at the largest angle is the solution.

For a general solution, one approach is to construct equilateral triangles on each side of the triangle (actually only two are needed) and draw the segments connecting the opposite vertices of the original triangle and the newly created equilateral vertices. They intersect in a point which is the solution. The point is called the Fermat point.
Because the construction is so similar to the one for Napoleon's Point, students often get the two confused. As a consequence of the construction it can be shown that the segments PA, PB, and PC are all at 120o angles to one another.
The Fermat point is also the common point of intersection of the circumcircles of the three equilateral triangles. The length of each of the three segments A-A', B-B', and C-C' is the same, and even more interesting is that their length is the same as the sum of the distances from the Fermat Point to the three vertices of the triangle.

Here are some comments on the history of the Fermat point by Harold W Kuhn from his post (edited somewhat) to the Historia Matematica list. Note that he refers to the point as the Torricelli point, which is common for mathematicians on the Continent I am told.
