Simple ratio

of three points $M_1,M,M_2$ on a straight line

The number $\lambda$ such that

$$\overline{M_1M}=\lambda\overline{MM_2}$$

One says, moreover, that $M$ divides the segment $M_1M_2$ in the ratio $\lambda$. If $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of $M_1$ and $M_2$, then the coordinates of $M$ are

$$x=\frac{x_1+\lambda x_2}{1+\lambda},\quad y=\frac{y_1+\lambda y_2}{1+\lambda}.$$

The simple ratio is an invariant of affine transformations.