Theorem

The centroid divides each median in the ratio $2 : 1$

Geometry • Triangle • Medians and Centroid

Formula

A G : G D = 2 : 1

In any triangle the three medians are concurrent at a single point

G

, the centroid, and

G

divides each median internally in the ratio

2 : 1

measured from the vertex: if

A D

is the median from

A

to the midpoint

D

B C

, then

A G : G D = 2 : 1

, so

A G = \frac{2}{3} A D

. Because the division is internal,

G

always lies strictly inside the triangle.

How to use it

1Read a median length off the centroid, or a centroid distance off a median, by taking $\frac{2}{3}$ or $\frac{1}{3}$ of it.
2Because $G$ is interior, $△ G A B$ , $△ GB C$ and $△ GC A$ are genuine triangles, so the triangle inequality can be applied to $\frac{2}{3}$ of the medians.
3The three of them have equal area, each $\frac{1}{3}$ of the whole triangle.
4'The centroid lies inside' is the property that separates it from the orthocentre and the circumcentre, which leave an obtuse triangle.

Worked example

△ P QR

the median

P M

measures

18

cm and

G

is the centroid. Find

P G

and

GM

Step-by-step

1
$P G : GM = 2 : 1$ , so $P G = \frac{2}{3} P M$ and $GM = \frac{1}{3} P M$
Why: The centroid cuts the median in $2 : 1$ from the vertex $P$ .
2
$P G = \frac{2}{3} \times 18 = 12$ cm, $GM = \frac{1}{3} \times 18 = 6$ cm

Answer

P G = 12

cm and

GM = 6

Watch out

The

2 : 1

is measured from the VERTEX, not from the midpoint — reading it the other way round halves every answer. And the concurrency point of the medians is the centroid alone: the altitudes meet at the orthocentre and the perpendicular bisectors at the circumcentre, and only the centroid is guaranteed to be interior.

Builds on

Midpoint theorem

Also known as

centroid, medians are concurrent, centroid ratio 2:1, point of intersection of the medians, G divides the median in 2:1, median concurrency

The centroid divides each median in the ratio $2 : 1$

Formula

Worked example

Appears in past papers

2019 (I)

2018 (I)