23 Among all triangles with a given perimeter, find the triangle with the largest inradius.

Among all triangles with a given perimeter, find the triangle with the largest inradius.
23. Let $a$, $b$, $c$ be the side lengths of a triangle with a fixed semiperimeter $p$, and let $S$ and $r$ be its area and the radius of its inscribed circle, respectively. Then, by the AM-GM inequality, we have $(r p)^{2}=S^{2}=p(p-a)(p-b)(p-c) \leqslant$ $p\left[\frac{(p-a)+(p-b)+(p-c)}{3}\right]^{3}=\frac{p^{4}}{27}$. This leads to $r \leqslant \frac{p}{\sqrt{27}}$, with equality if and only if $p-a=p-b=p-c$, i.e., when the triangle is equilateral, $r$ attains its maximum value.