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if you have a line segment and you know the coordinates of the two endpoints

how do you find the coordinates of the point a third of the way along the line?

In general how do find the coordinates of the point m/n along the line where m and n are positive integers

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Suppose $p=(p_1, p_2), q=(q_1, q_2)$. Points along the segment between are defined (for $0\le \lambda \le 1$) as $$\lambda p + (1-\lambda)q=(\lambda p_1 + (1-\lambda)q_1, \lambda p_2+(1-\lambda) q_2)$$

To be two-thirds of the way along from $q$ to $p$, choose $\lambda=\frac{2}{3}$.

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