Centroid of a Triangle

Properties of the Segment Connecting the Midpoints of Two Sides of a Triangle

In , let and be the midpoints of and , respectively. Then:
Thus, .
From the property of line segments between parallel lines in a triangle:
Therefore, .
In summary, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and has a length equal to half the length of the third side.

Centroid of a Triangle

In a triangle, the segment connecting a vertex to the midpoint of its opposite side is called a median. A triangle has three medians.
In , let be the intersection point of two medians, and .
Since and are the midpoints of and , respectively, we have:
Thus, , and the similarity ratio is:
This implies:

Similarly, considering another median , let be its intersection with . By the same reasoning:
Since both and divide in a ratio, they coincide ( ).
Therefore, the three medians of intersect at a single point, , which divides each median in a ratio, measured from the vertex to the midpoint of the opposite side. This point is called the centroid of the triangle.

Summary of Centroid Properties

  1. The three medians of a triangle intersect at a single point called the centroid.
  2. The centroid divides each median in a ratio, measured from the vertex to the midpoint of the opposite side.
    For :