Section Formula — Formula Sheet

x=\frac{m x_{2}+n x_{1}}{m+n}, y=\frac{m y_{2}+n y_{1}}{m+n}

\frac{m}{n}=\frac{x-x_{1}}{x_{2}-x}=\frac{y-y_{1}}{y_{2}-y}

\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)

x=\frac{x_{1}+x_{2}+x_{3}}{3}, y=\frac{y_{1}+y_{2}+y_{3}}{3}

D=\left(\frac{x_{2}+x_{3}}{2}, \frac{y_{2}+y_{3}}{2}\right)

x=\frac{2 \times\left(\frac{x_{2}+x_{3}}{2}\right)+1 \times x_{1}}{2+1}=\frac{x_{1}+x_{2}+x_{3}}{3}

y=\frac{2 \times\left(\frac{y_{2}+y_{3}}{2}\right)+1 \times y_{1}}{2+1}=\frac{y_{1}+y_{2}+y_{3}}{3}

If A(x1,y1) and B(x2,y2) are given points and P divides AB internally in the ratio m:n, then P is given by x=(mx2+nx1)/(m+n), y=(my2+ny1)/(m+n).

From similarity, m/n=(x-x1)/(x2-x)=(y-y1)/(y2-y).