IMO 1968 SL 7

Origin: HUN

Problem

Prove that the product of the radii of three circles exscribed to a given triangle does not exceed $3\sqrt{3}$ times the product of the side lengths of the triangle. When does equality hold?

Solution

Let $r_a$, $r_b$, $r_c$ denote the radii of the exscribed circles corresponding to the sides of lengths $a$, $b$, $c$ respectively, and $R$, $p$ and $S$ denote the circumradius, semiperimeter, and area of the given triangle. It is well-known that

$$r_a(p-a)=r_b(p-b)=r_c(p-c)=S= \sqrt{p(p-a)(p-b)(p-c)}=\frac{abc}{4R}.$$

Hence, the desired inequality $r_ar_br_c \leq 3\sqrt{3},abc$ reduces to $p \leq 3\sqrt{3},R$, which is by the law of sines equivalent to

$$\sin \alpha + \sin \beta + \sin \gamma \leq \frac{3\sqrt{3}}{2}.$$

This inequality immediately follows from Jensen’s inequality, since the sine is concave on $[0,\pi]$. Equality holds if and only if the triangle is equilateral.