Medians of a Triangle

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 length of a median can be found from the lengths of the three original sides of the triangle. If the side lengths are a, b, and c, then the length of the median from vertex A to side a is given by

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.