Task 15. What does Heron's formula for a triangle look like, where the sides are expressed as irrational numbers?
Solution. According to Heron's formula

$$
16 S^{2}=(a+b+c)(b+c-a)(a+c-b)(b-c+a)
$$

or

$$
\begin{gathered}
16 S^{2}=\left((a+c)^{2}-b^{2}\right) \cdot\left(b^{2}-(c-a)^{2}\right)= \\
\left(a^{2}+2 a c+c^{2}-b^{2}\right) \cdot\left(b^{2}-c^{2}+2 a c-a^{2}\right)
\end{gathered}
$$

or

$$
16 S^{2}=(2 a c)^{2}-\left(a^{2}+c^{2}-b^{2}\right)^{2}
$$

or

$$
S=\frac{1}{4} \sqrt{4 a^{2} c^{2}-\left(a^{2}+c^{2}-b^{2}\right)^{2}}
$$