Application of Modern Dynamic Software to Fermat's Problem

Application of Modern Dynamic Software to Fermat’s
Problem
Jordan Tabov
(joint work with Nevena Koleva)
The Fermat Problem is not only challenging but gives good opportunities for application
of different classical and modern ideas and techniques and therefore is suitable
for the work with talented students. Here we show how it can be reduced to a relatively
simple problem involving angles, a problem, which can be investigated by
some modern dynamic software like GeoGebra, and even can be used for demonstrating
the power of the modern modeling computer tools in geometrical research.
Recall that the general Fermat Problem can be formulated in the following way.
Given a triangle ABC and real numbers l, m and n, determine the minimal value of
the function F (X) = l · XA + m · XB + n · XC, where X is a variable point which
ranges in the entire plane of the triangle ABC.
When the coefficients l, m and n are positive numbers, some of the most efficient and
beautiful methods for attacking it - using respectively spiral homothety, circumscribed
triangle, Ptolemy’s Theorem, complex numbers, mechanical approach - are presented
in many articles and books, of which here we quote some of the most interesting and
useful ones: Jalal and Krarup [1], Shay and Tessler [2], Hofmann [3], Courant and
Robbins [4], Ganchev and Nikolov [5].
In the present paper we consider the case when l, m and −n (n < 0) are positive
numbers. The Fermat Problem with one negative and two positive coefficients can be
reduced to the case with positive coefficients, as it is done in [5], or can be solved using
methods of classical geometry (as in [1]). Here we apply a functional approach (the
idea of this approach appeared in Tabov [6]).
Keywords: Fermat Problem, dynamic software.
References
[1] G. Jalal, J. Krarup, Geometrical solutions to the Fermat problem with arbitrary
weights, Annals of Operations Research 123 (2003), 67–104.
[2] G. Shay, R. Tessler, The Fermat-Steiner Problem, Am. Math. Monthly 109
(2002),
443–451
[3] E. Hofmann, Elementare Losung einer minimumsaufgabe, Zeitschrift fur mathematischen
und naturwissenschaftlichen Unterricht, 60 (1929), 22–23
[4] Courant, R., H. Robbins. What is Mathematics? Oxford University Press, New
1

2
York, 1951
[5] G.Ganchev, N. Nikolov, Isogonally conjugated points and Fermat’s Problem,
Matematicheskoe prosveshchenie, 12 (2008), 185–194. (In Russian)
[6] J. Tabov, Solution to Problem 866 (proposed by J. Dou), Crux Mathematicorum,
10 (1984), 327–329
Institute of Mathematics and Informatics Bulgarian Academy of Sciences