How long are the medians of a triangle if the sides of the triangle are given?
Let the midpoints of the sides of triangle $ABC$ be denoted by $A_{1}$, $B_{1}$, and $C_{1}$ respectively. The quadrilateral $AB_{1}A_{1}C_{1}$ is a parallelogram, so according to the previous problem,

![](https://cdn.mathpix.com/cropped/2024_05_02_38e00b76a9b2abb5fa14g-1.jpg?height=79&width=546&top_left_y=237&top_left_x=779)

For the sake of brevity, let

$$
A A_{1}=s_{a}, \quad B C=a, \quad C A=b \quad \text { and } \quad A B=c,
$$

then

$$
s_{a}^{2}+\frac{a^{2}}{4}=2\left(\frac{c^{2}}{4}+\frac{b^{2}}{4}\right)
$$

or

$$
s_{a}=\frac{1}{2} \sqrt{2 b^{2}+2 c^{2}-a^{2}}
$$

Similarly,

$$
s_{b}=\frac{1}{2} \sqrt{2 c^{2}+2 a^{2}-b^{2}}
$$

and

$$
s_{c}=\frac{1}{2} \sqrt{2 a^{2}+2 b^{2}-c^{2}}
$$

(Izidor Szóbel, Beregszász.)