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.


"Our story starts with a problem rather casually posed by Fermat early in the 17th century. At the end of his celebrated essay on maxima and minima, in which he presented pre-calculus rules for finding tangents to a variety of curves, he threw out the challenge: "Let he who does not approve of my method attempt the solution of the following problem: Given three points in the plane, find a fourth point such that the sum of its distances to the three given points is a minimum!" The problemmay have travelled to Italy with Mersenne; it is known that before 1640 Torricelli had solved the problem. He asserted that the circles circumscribing the equilateral triangles constructed on the sides of and outside the given triangle intersect in the point that is sought. This point is called the Torricelli point. Also, in Cavalieri's "Exercitationes geometricae" of 1647, it is shown that the sides of the given triangle subtend angles of 120 degrees from the Torricelli point. Furthermore, Simpson asserted and proved in his "Doctrine and Application of Fluxions" (London, 1750) that the three lines joining the outside vertices of the equilateral triangles defined above to the opposite vertices of the given triangle intersect in the Torricelli point. These three lines are called Simpson lines.
I cannot end this historical sketch without mention of the fact that the Fermat problem has been widely popularized by Courant and Robbins (in "What is Mathematics?") under the name of the "Steiner Problem". Although this gifted geometer of the 19th century can be counted among the dozens of mathematicians who have written on the subject, he does not seem to have contributed anything new, either to its formulation or its solution. As for the statement by Courant and Robbins that the generalization of the problem to more than three points is a sterile generalization, their answer is found in the recent literature, which has added new applications and understanding through this "sterile" extension of the problem.