The Fermat-Torricelli point is one of the many centers of a triangle.
How can we calculate one using geogebra?
Let as look at the triangle that has vertices (0,0), (3,0), (0,4).
1) Draw those three points on the screen.
2) Label the points A,B and C respectively.
3) Use the polygon tool to form triangle ABC.
4) Use the regular polygon tool and click points C and then B and enter 3 in the input box.
5) Another triangle will be created with BC as the base. It will have apex D.
6) Use the line tool and draw a line through D and A.
7) Use the circle through three points tool and click D, C, and B.
8) A circle will be created.
9) Using the intersection tool get the intersection of that circle and the line through A and D.
10) The point F will be created and it marks the spot.
11) Set rounding to 15 and read off the value of F in the algebra pane. That is the Fermat-Torricelli point.
Now using the techniques of experimental mathematics can we conjecture a closed form for the point. Yes we can!
In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.