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I am trying to find the coordinates of a middle point of a line in $\mathbb{R}^n$. Let $X(x_1, \ldots, x_n)$ and $Y=(y_1, \ldots, y_n)$ be two points in $\mathbb{R}^n$.

How do I find the middle point $Z$ on the line $XY$?

Thank you.

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Do you know how to find the coordinates of the midpoint of the line segment from $(3, 10)$ to $(14, 6)$ in $\mathbb R^2$? – Jonas Meyer Sep 26 '12 at 5:03
up vote 2 down vote accepted

The midpoint of the segment from $a$ to $b$ on the real line is simply the arithmetic mean (ordinary average) of $a$ and $b$, $\frac12(a+b)$. The midpoint of the segment from $\langle a_1,a_2\rangle$ to $\langle b_1,b_2\rangle$ in the plane is $$\left\langle\frac{a_1+b_1}2,\frac{a_2+b_2}2\right\rangle\;,$$ found by averaging the coordinates; the proof is a simple argument involving similar triangles. Now generalize. You want the point that is midway between $X$ and $Y$ in every coordinate direction.

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