Students define a median

Ninth grade students using a dynamic geometry program are asked to explore possible definitions of median in the triangle formed by the white dots. When the students choose a vertex and ask the program to produce a median, the program responds by drawing the white line and the gold dot (see below). Students explore the values of areas, angles and side lengths and suggest the following “definitions” – A – a median divides the side opposite the vertex in two equal pieces B – a median divides the area of the triangle into two equal areas Can you show that these definitions are equivalent – if either one is true then the other is as well? Why might definition A be preferable to definition B ? Why might definition B be preferable to definition A ? Can the definition of median be generalized to polygons with more than three sides?