In this article, we consider certain minimization problems. If d[subscript 1], d[subscript 2] and d[subscript 3] are the distances of a boundary or inner point to the sides of a given triangle, find the point which minimizes d[subscript 1][superscript n] + d[subscript 2][superscript n] + d[subscript 3][superscript 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[subscript 1][superscript n] + d[subscript 2][superscript n] + d[subscript 3][superscript n], n [greater than or equal to] 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'.