The medians of a triangle are segments
from each vertex to the midpoint of the opposite side. The medians always
intersect in a single point, called the centroid,
or geocenter. Proof of this property is an easy corollary to Ceva's
Theorem.
In the figure, the medians are shown
in red, and the centroid is point G (a common mathematical abbreviation
for the geocenter). It is also easy to show that the segment from G to
a vertex is always twice as long as the segment from G to the midpoint
of the opposite side. The Geocenter is on the Euler line one third of the
way from the circumcenter to the orthocenter.
The lines connecting the
three medians divide the triangle into four smaller congruent triangles.
The triangle connecting the three medians is similar to the original triangle
with sides 1/2 as long and rotated 180o about the geocenter.
The midpoints of the triangle are on the nine-point circle, which is the
circumscribing circle of the triangle joining the midpoints. If a triangle
were made from a uniformly dense material, the intersection of the medians
would be the point where the triangle would balance on a pin point. This
was known to Archimedes around 250 BC.
If you make a second triangle with three sides that are the lengths of the medians of triangle ABC, the new triangle will have an area of 3/4 the area of ABC. You can click here to see an interactive Java sketch of a simple visual proof of this theorem.
The lengths of the medians and sides
of a triangle are also related by the fact that the sum of the squares
of the medians is 3/4 the sum of the squares of the sides. ma2+mb2+mc2=(3/4)(a2+b2+c2). The triangle formed by the line segments joining the three medians of a triangle is called the medial triangle. 
An angle bisector of any vertex of the medial triangle (ray FP in figure) will divide the perimeter of the original triangle into two equal parts. The incenter of the medial triangle is called the Spieker point. It is the center of gravity of a homogeneous wire frame triangle shaped like ABC. The line joining the Spieker point, and the Geocenter also passes through the Incenter and the Nagel point of the original triangle.