Centroid of a Triangle

Definition

The centroid (or barycenter) of a triangle is the point where the three medians of the triangle intersect. It represents the “center of gravity” of a triangle with uniform density.

Why It Matters

It identifies the point of perfect balance in a triangular structure, which is a fundamental requirement for stability in architecture and mechanical design.

Core Concepts

• Median: A line segment connecting a vertex of a triangle to the midpoint of the opposite side.
• Location Property (Theorem 7.2.4): The centroid $C$ is located two-thirds of the way from each vertex to the midpoint of the opposite side. If $RM$ is a median, then:
  • $RC = \frac{2}{3}(RM)$
    • How to read: “The segment R C equals two thirds times the segment R M.”
    • Meaning: From vertex $R$ to centroid $C$ is exactly two-thirds of the full median length—the centroid sits closer to the vertex than to the midpoint.
  • $CM = \frac{1}{3}(RM)$
    • How to read: “The segment C M equals one third times the segment R M.”
    • Meaning: From centroid $C$ to midpoint $M$ is the remaining one-third; together $RC + CM = RM$.
• Coordinate Formula: If the vertices are $(x_1, y_1), (x_2, y_2),$ and $(x_3, y_3)$, the centroid $C$ is: $C = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)$
  • How to read: “The centroid C equals the point with coordinates x one plus x two plus x three all divided by three, and y one plus y two plus y three all divided by three.”
  • Meaning / when to use: The centroid is the arithmetic average of the vertex coordinates. Use this when vertices are given numerically and you need the balance point without drawing medians.

Connected Concepts

• Points of Concurrence in a Triangle
• Special Segments in Triangles
• Triangle Classification
• Distance Formula
• Midpoint Formula