An applet to show that the three medians of a triangle all meet in a
single point. The median of a triangle is a line from one
vertex to the midpoint of the opposite side.

The three vertices of the triangle above can each be dragged to show
that
the three medians still meet in a single point when we look at other
triangles

When a mathematician sees a behavior like this that works with all
triangles
(or at least with a bunch of examples that we have looked at) the
suspicion is that there must be a structure that helps us prove that it
must always happen.

For this construction we first want to
establish a lemma, that if we take three copies of a triangle, two in a
row and the third flipped to fill in, that the base line forms a
straight line. This uses the fact that the sum of the angles of a
triangle adds up to 180 degrees, which is the angle of a straight
line.
The lemma check box gives us this construction, with the point n
draggable. In particular, note that the point Q stays on the
median for all locations of N.

From this we build to a final construction, putting 9 small
triangles together into a single large triangle with each side three
times as long. consider the center point of any line parallel to
an edge. That point must lie on the corresponding median.
Thus the central point of the construction must lie on all three
medians.

Created with GeoGebra
GeoGebra is a GNUed software package for mathematics visualization.
The home for the applications is http://www.geogebra.org.