Median of a triangle

A median of a triangle is a cevian of the triangle that joins one vertex to the midpoint of the opposite side.

In the following figure, $AM$ is a median of triangle $ABC$ .

Each triangle has $3$ medians. The medians are concurrent at the centroid. The centroid divides the medians (segments) in a $2:1$ ratio.

Stewart's Theorem applied to the case $m=n$ , gives the length of the median to side $BC$ equal to

$\frac 12 \sqrt{2AB^2+2AC^2-BC^2}$

This formula is particularly useful when $\angle CAB$ is right, as by the Pythagorean Theorem we find that $BM=AM=CM$ .

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