Minimizing inside a triangle with GeoGebra

ABSTRACT

In this article, we consider certain minimization problems. If $d_1,d_2$ and $d_3$ are the distances of a boundary or inner point to the sides of a given triangle, find the point which minimizes $d_1^n+d_2^n+d_3^n,$ for positive integer n. These problems can be afforded easily with GeoGebra. We consider two examples, the first concerns an isosceles triangle and the second a scalene triangle. In both cases, we divide the triangle horizontally into line segments parallel to the base and look at a family of polynomial functions. Using GeoGebra, we observe in the case of isosceles triangle that the minimum point of each member of the family lies on the y-axis. Tracking these points vertically we discover the critical point which minimizes $d_1^n+d_2^n+d_3^n$, $n\ge 2.$ In particular, we show that the sequence of these critical points converges to the incenter of the triangle. In the case of a scalene, we observe that the minima points of the polynomials lie on a curve, the minimum of which can be traced with GeoGebra and computed with basic calculus. Finally, we consider a discussion with some references concerning general solutions of ‘minimal sums of distances’ and ‘minimal sums of squared distances’.

KEYWORDS:

• Minimization problem
• distance
• triangle
• GeoGebra

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 51N20
• 97M50

Acknowledgments

The author is indebted to the reviewers for their comments and valuable suggestions concerning the paper.

Disclosure statement

The author declares that there is no potential conflict of interest.