Overview

In a triangle, a median is a segment from a vertex to the midpoint of the opposite side. Medians intersect at the centroid, which splits each median in a $2:1$ ratio. This point is also the balance point of the triangle. They also give rise to triangles with equal areas.

Definitions and Key Ideas

• The midpoint of a segment is the point that divides the segment into two equal lengths.
• A median connects a vertex to the midpoint of the opposite side.
• In a triangle, the three medians intersect at a single point called the centroid.
• The centroid divides each median $2:1$ from the vertex.
• The three medians divide the triangle into 6 equal-area small triangles.
• Each median splits the triangle into two equal-area triangles.
• The centroid's coordinates are the average of the coordinates of the three vertices.

Area Properties

In $\triangle ABC,$

and similarly for the other three "half" triangles.

Also, the six triangles the median splits the larger triangle into have equal area. In other words,

Proof: If we drop an altitude from $A$ to $BC,$ triangles $AMB$ and $AMC$ will share this altitude. Their bases are also the same length since $BM=MC.$ Then using $Area = \frac{1}{2} bh,$ these triangles have the same area. Also, if we drop an altitude from $G$ to $BC,$ we can show $[GBM] = [GMC] = x.$ Repeat for the other three pairs of triangles, obtaining $[GLA] = [GBL] = y$ and $[GAN] = [GNC] = z.$ Then from our previous result, $2y+x = 2z+x.$ Thus $y=z.$ By a similar argument, $x=y,$ so $x=y=z$ and we are done.

Ratio Properties

In $\triangle ABC,$ with $M$ as the midpoint of $BC$ and $G$ the centroid, it holds that

$$AG : GM = 2 : 1.$

Proof: We follow the opposite reasoning as our area properties proof. We know that $[BAG] = 2[BGM].$ Therefore since these triangles share an altitude from $B$ to $AM,$ $AG = 2GM.$ Repeat for the other three medians symmetrically.

Connection to Coordinate Geometry

In a triangle with vertices $(x_1, y_1), (x_2,y_2), (x_3,y_3),$ the centroid has coordinates

You may recognize this as the average of the corresponding $x$ and $y$ coordinates.

Proof: First, the midpoint of $(x_2, y_2), (x_3,y_3)$ is

Next, the point $2/3$ of the way to this point along the median from $(x_1,y_1)$ will be

Common Pitfalls

• Flipping the $2:1$ ratio, or mistaking it for $1:1.$
• Assuming medians are perpendicular from their appearance on a diagram.
• Confusing medians with angle bisectors or altitudes.

Worked Example

In triangle $ABC$, the centroid $G$ lies on median $AM$. If $AG=10$, find $GM$.

The centroid ratio is $AG:GM = 2:1$, so $GM = 5$.

More Examples

Example 1: Full Median Length

If $AG=8$ on median $AM$, find $AM$.

$AG = \frac{2}{3}AM$, so $AM = 12$.

Example 2: Area Split

In triangle $ABC$, median $AM$ splits the triangle into two triangles of equal area. If $[ABC]=42$, find $[ABM]$.

$[ABM]=21$.

Example 3: Six Equal Areas

The medians divide a triangle into six equal areas. If one of the small triangles has area $5$, what is the area of the whole triangle?

$6\cdot 5 = 30$.

Practice Problems