If a point P divides the line segment joining A (position vector a) and B (position vector b) in the ratio m:n, then: Internal division: p = (mb + na)/(m+n). External division: p = (mb - na)/(m-n). Special cases: Midpoint (m=n=1): p = (a+b)/2. Centroid of triangle with vertices a, b, c: g = (a+b+c)/3. Incenter: r = (a·A + b·B + c·C)/(a+b+c) where a, b, c are side lengths opposite to vertices A, B, C.