Problem 18. Among all triangles of a given perimeter, find the one that has the maximum area.
Solution. According to Heron's formula

$$
S^{2}=p \cdot(p-a) \cdot(p-b) \cdot(p-c)
$$

and by the condition of the problem $p=\frac{a+b+c}{2}=$ const. Note that:

$$
(p-a)+(p-b)+(p-c)=3 p-2 p=p=\text { const. }
$$

It is known that the product of three positive factors is maximized when they are equal, given a constant sum, i.e.

$$
p-a=p-b=p-c, \text { or } a=b=c
$$

Answer. The triangle with the maximum area for a given perimeter is the equilateral triangle.