Task B-3.2. The length of one height of a triangle is equal to the sum of the lengths of the remaining two heights. Write the length of the shortest side of the triangle using the lengths of the remaining two sides.
## First solution.

Let $v_{a}=v_{b}+v_{c}$. Then the shortest side of the triangle is side $a$.

Multiply the given equality by $\mathrm{s} \frac{a b c}{2}$.

$$
v_{a} \cdot \frac{a b c}{2}=v_{b} \cdot \frac{a b c}{2}+v_{c} \cdot \frac{a b c}{2}
$$

If we let $P$ be the area of the triangle, we get that $P \cdot b \cdot c=P \cdot a \cdot c+P \cdot a \cdot b$.

It follows that

$$
a=\frac{b c}{b+c}
$$