The origin of the Fermat–Weber problem is attributed to the great Pierre de Fermat (1601–1665) who asked Evangelista Torricelli (1608–1647) to find the point

that minimizes the sum of the Euclidean distances to three given points

,

,

in the plane. Around 1640, Torricelli devised a geometrical construction for this problem. He showed that

subtends an angle of

to each of the sides of the triangle

. However, this is true only when each of the interior angles of the triangle is less than

. The so-called

*complementary problem,* where one angle of the triangle can be greater than

first appeared in [1]. Its solution states that the optimum point always coincides with the obtuse vertex of the triangle; this was correctly proved later [2].

In this Demonstration we show how the position of the Fermat–Weber point for

varies with

and

. It is observed that for every odd positive integer

, there exists an integer

such that whenever

, the Fermat–Weber point of

coincides with the vertex of

lying on the

* *axis. This can be thought of as an extension of Courant and Robbins' complementary problem on triangles as described above. The proof of this fact can be found in [3], where it is also shown that

, when

(

) is any odd positive integer.

The values of the quantities

* *and

* *can be controlled using their respective sliders. For every fixed value of

*, *the control for the values of

can be set to autorun to observe the movement of the Fermat–Weber point of the chain, as described above. Note that the range of

in the Demonstration is set up to 21. The range can be increased by changing the range of the outer

Manipulate function in the code.

[1] R. Courant and H. Robbins,

*What Is Mathematics?*, New York: Oxford University Press, 1996.

[2] J. Krarup and S. Vajda, "On Torricelli's Geometrical Solution to a Problem of Fermat,"

*IMA Journal of Mathematics Applied in Business and Industry*,

**8**, 1997 pp. 215–224.

[3] B. B. Bhattacharya, "On the Fermat–Weber Point of a Polygonal Chain and Its Generalizations,"

*Fundamenta Informaticae*, to appear. (accepted October 2010)