I read from http://rgmia.org/papers/v6n3/wsh.pdf the following: A triangle with semiperimeter $s$, circumradius $R$ and inradius $r$ satisfies $$16Rr-5r^2\leq s^2\leq 4R^2+4Rr+3r^2.$$ How can I prove it, or what is the sketch of the proof?
2026-04-13 05:51:19.1776059479
How one can show Gerretsen's inequality?
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The idea is to fix $R$ and $r$ and try to maximize and minimize $s$.
It can be shown that amongst triangles with inradius $r$ and circumradius $R$ the value of $s$ is maximal/minimal for isosceles triangles. I think it's quite nontrivial fact and I don't see how to prove it geometrically. It seems that Poncelet porism and the formula for distance between circumcenter and incenter might be useful.
So you only need to consider these two particular triangles maximizing and minimizing $s$. You can easily express the sides of the triangle in terms of $r$ and $R$, then you can divide the inequality by $R^2$ and you are left with inequality containing only one variable $0 < t=\frac rR \le \frac 12$. Some easy calculus should lead to the end of the proof.