A proposal to explore geometry with GeoGebra: graphical exploration and formal demonstration of the collinearity of the barycenters of a polygon

This paper illustrates an example of a mathematical activity that teachers and students can replicate to create an experience that resembles professional mathematical activity. We extend the property “Consider a triangle ABC, any straight line and let A’, B’, C’ be the reflections of the points A, B, C on the straight line then the barycenters of the triangles ABC, A’BC, AB’C and ABC’ are collinear and the line of collinearity is perpendicular to the straight line” for any quadrilateral. It is proved that there are four additional triangles, for a total of eight, whose barycenters are collinear and that there are eleven additional quadrilaterals. Some of the geometric and numerical experiments in which GeoGebra software was used, necessary to extend and demonstrate analogous results for the case of a pentagon, are described, and the corresponding results for an n-sided polygon are generalized and demonstrated.