Centroid of a Triangle
Lesson · Intermediate
Geometry: Plane Geometry
The centroid of a triangle is the point where its three medians intersect.
A median connects a vertex to the midpoint of the opposite side.
If G is the centroid of △ABC and M is the midpoint of BC, then
Equivalently,
Properties
1. The Three Medians Are Concurrent
The three medians of a triangle intersect at a single point, the centroid.
2. The Medians Divide the Triangle into Six Equal-Area Triangles
If all three medians are drawn, the six small triangles have equal areas.
In particular, each has area
3. The Centroid Divides Each Median in a 2:1 Ratio
The longer part is always the part from the vertex to the centroid.
4. Coordinate Formula
If
then the centroid is
Let G be the intersection of medians AM and BN. We want to prove CL is a median, too.
By Toolkit: Ratios of Areas
Similarly, BN is a median
By (1), (2)
Let
Similarly, let
Similarly
By Toolkit: Section Formula
In the figure below, G is the centroid of △ABC. Points X and Y lie on BC, with GX ∥ AB and GY ∥ AC. Prove that
Draw median AM.
GX ∥ AB. By Toolkit: Triangle Proportionality Theorem
Similarly
So,
G is the centroid of △ABC, and ℓ is a line through G. The line ℓ intersects AB, AC, and the extension of BC at Z, Y, and X, respectively. Prove that
Draw three perpendiculars from A, B, and C to ℓ and call them x, y, and z.
Multiplying by 2
Dividing by GY,