# Steiner Point Construction

This just in from Mrs. Hill:

Topics in Discrete Mathematics students got to review some lessons from geometry when they learned how to construct the Steiner Point of a triangle. The Steiner Point in a triangle is the point from which three branches lead out to the triangle’s vertices at perfect 120° angles. (See diagram below.) This Steiner Point has important modern day applications for creating the shortest (and therefore cheapest) possible fiber optic network between any three locations. If a triangle’s angle measures are all less than 120°, then the Steiner Point can be found inside the triangle using a geometric construction first developed by Italian mathematician Evangelista Torricelli in the early 1600’s. Torricelli (most famous for his work in physics but also an accomplished mathematician) was a protégé of Galileo, and in 1641 succeeded Galileo as the court mathematician to Grand Duke Ferdinand II of Tuscany.

Torricelli’s method for finding the Steiner Point in a triangle requires only the use of the traditional geometric construction tools – straightedge and compass. Using properties of equilateral triangles and inscribed angles from elementary Euclidean Geometry, the basics of the construction are shown below in Figure 1 (a,b,c). The students in Topics further learned how the Steiner Point was used in 1989 to connect Hawaii, Japan, and Guam via the Third Trans-Pacific Cable (TPC-3). Also below, Figure 2 shows a stamp issued by the Japanese Postal Service to commemorate the completion of TCP-3. As one can see from the picture, the three undersea cables meet at 120° angles under the western Pacific Ocean.

Student constructions: