Notes - Centroid of a triangle and triangle midsegment theorem

The centroid of a triangle is the point where all three medians intersect. A median is a line segment from a vertex to the midpoint of the opposite side. The centroid has several important properties:

• Location: The centroid is located inside the triangle for all types of triangles (acute, right, and obtuse).

• Ratio: The centroid divides each median into two segments with a ratio of 2:1, with the longer segment being closer to the vertex.

• Coordinates: If a triangle has vertices at points
  $(x<em>1,y</em>1)$,
  $(x<em>2,y</em>2)$, and
  $(x<em>3,y</em>3)$, then the coordinates of the centroid
  $(G)$ can be found using the formula:
  Gigg(xG,yGigg) = igg(\frac{x1+x2+x3}{3},\frac{y1+y2+y3}{3} \bigg).

Triangle Midsegment Theorem

The Triangle Midsegment Theorem states that a line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half the length of that side. This theorem provides important relationships between the sides of a triangle:

• Midsegment Properties: If a triangle has vertices at
  $(A, B, C)$, and midpoints
  $(D, E)$ on sides
  $(AB)$ and
  $(AC)$ respectively, then the segment
  $(DE)$ that connects the midpoints is parallel to side
  $(BC)$ and:
  $DE = \frac{1}{2} BC$.

• Application: This theorem is useful in various geometric proofs and constructions, particularly in establishing relationships between different parts of a triangle and solving for lengths or proportions.