Geometry • Topic 8

Centroid

The centroid ( $G$ ) is the "center of mass" of a triangle. It is formed by the intersection of the three medians.

Definition

Definition (Median)

A median of a triangle connects a vertex to the midpoint of the opposite side.

Theorem (Concurrency)

The three medians of any triangle are concurrent at a single point called the centroid (

G

The centroid divides each median in a

2 : 1

ratio. The longer segment is always on the vertex side.

A G = 2 G M_{a}

where

M_{a}

is the midpoint of

BC

If vertices are

A (x_{1}, y_{1})

B (x_{2}, y_{2})

C (x_{3}, y_{3})

, the centroid is simply the average:

G = (\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3})

Exercise (Problem 1)

△ A BC

, medians

A D

and

BE

are perpendicular. If

A C = 13

and

BC = 10

, find the length of

A B

Exercise (Problem 2)

Prove that for any point

P

and centroid

G

△ A BC

P A^{2} + P B^{2} + P C^{2} = G A^{2} + G B^{2} + G C^{2} + 3 P G^{2}

(Leibniz's Theorem).

Exercise (Problem 3)

Can the centroid, orthocenter, and circumcenter of a triangle form a non-degenerate triangle? (Think about the Euler Line).