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Suppose you have two points $p_1 = x_1:y_1:z_1$ and $p_2 = x_2:y_2:z_2$ in trilinear coordinates. If $p_1 = (a_1,b_1,c_1)$ and $p_2 = (a_2,b_2,c_2)$ in actual directed distances, then the midpoint of the segment $p_1p_2$ is obviously $\left( \frac{a_1 + a_2}{2} , \frac{b_1 + b_2}{2} , \frac{c_1 + c_2}{2} \right)$, by considering the right trapezoids formed when dropping the perpendiculars.

However, this fails when considering the proportional coordinates. How can you calculate the midpoint of a segment if you are not given the absolute trilinear coordinates of the endpoints?

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No problem at all. If you´re given only the proportional coordinates of a point, nothing prevents you from calculating the absolute trilinear coordinates.

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