Medians of Triangle

In a triangle, a median is a line joining a vertex with the mid-point of the opposite side.
Every triangle have 3 medians.

The three medians meet at one point called centroid - point G.

medians of triangle
Here the medians are AX, BY, CZ and they meet at G.

The G point separates each into segments in ratio 2 : 1 i.e.:

$\frac{\overline{AG}}{\overline{GX}} = \frac{\overline{BG}}{\overline{GY}} = \frac{\overline{CG}}{\overline{GZ}} = \frac21$


$\frac{\overline{AG}}{\overline{AX}} = \frac{\overline{BG}}{\overline{BY}} = \frac{\overline{CG}}{\overline{CZ}} = \frac23$


$\frac{\overline{GX}}{\overline{AX}} = \frac{\overline{GY}}{\overline{BY}} = \frac{\overline{GZ}}{\overline{CZ}} = \frac13$

Median length formulas


Let's denote the medians by ma, mb, mc and the triangle sides by a, b, c.

$m_a = \frac{1}{2}\sqrt{2c^2+2b^2-a^2}$
$m_b = \frac{1}{2}\sqrt{2c^2+2a^2-b^2}$
$m_c = \frac{1}{2}\sqrt{2a^2+2b^2-c^2}$
Here are the formulas for calculating sides of a triangle when we have medians lengths.
$a = \frac{2}{3}\sqrt{2m_b^2+2m_c^2-m_a^2}$
$b = \frac{2}{3}\sqrt{2m_c^2+2m_a^2-m_b^2}$
$c = \frac{2}{3}\sqrt{2m_a^2+2m_b^2-m_c^2}$

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