### Steiner's Problem

This problem was named the Steiner problem for the first time in 1941, in the popular book *What is mathematics?*, by R. Courant and H. Robbins.

However, long before Steiner, other mathematicians like Fermat (1601 - 1665), Torricelli, Cavalieri and Simpson had approached the first variants of this problem.

It all started with a simple problem proposed by Fermat:

Torricelli found a solution to this problem in the case in which the internal angles of the triangle formed by the three given points \(A\), \(B\) and \(C\) are all smaller than or equal to 120º.

TORRICELLI's METHOD:

*Torricelli point*and it solves Fermat's problem.

When one of the inner angles of the triangle \([ABC]\), for instance the angle corresponding to vertex \(A\), is greater than or equal to 120º, the solution to Fermat's problem is unique and coincides with point \(A\). (Note that in such cases the Torricelli point is outside the triangle \([ABC]\) and thus it cannot be a solution to Fermat's problem.)

Cavalieri found an important property of the Torricelli point, which was published in his book *Exercitationes Geometricae* in 1647:

Later, Simpson found another way of constructing the Torricelli point and published it in his book *Doctrine and Application of Fluctions* in 1750.

SIMPSON's METHOD:

*Simpson segments*.

Also this method only solves Fermat's problem when the three internal angles of the triangle \([ABC]\) are smaller than or equal to 120º.

Another important property, mentioned by Heinen in 1834, is the following:

Steiner was attracted to this problem while trying to solve a Fermat's problem generalization exercise, proposed by Simpson in his book *Fluxions*:

Another variant of the problem is the proposal by Jarnik and Kössler in 1934:

In this section you can find pages with *sketches* about this problem.