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
  (x1,y1),
  (x2,y2), and
  (x3,y3), 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.