Internal and External Division Points of a Line Segment in Coordinate Space

Given two points and in coordinate space:

1. The coordinates of the point that divides the line segment internally in the ratio are given by:
2. The coordinates of the point that divides the line segment externally in the ratio are given by:

Midpoint and Centroid

1. The coordinates of the midpoint of the line segment formed by two points and are:
2. The coordinates of the centroid of the triangle , where the vertices are , , and , are: