The Garfunkel-Bankoff inequality and the Finsler-Hadwiger inequality (Q6162795)

In this short note, the following very interesting refinement of the Garfunkel-Bankoff inequality \[ \begin{aligned} \tan^2\frac{A}{2}+\tan^2\frac{B}{2}+\tan^2\frac{C}{2} \geq 2-8\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}+\frac{r^2(R-2r)}{4R^2(R-r)} \end{aligned} \] and the sharpening of the Finsler-Hadwiger inequality \[ \begin{aligned} a^2+b^2+c^2 \geq 4S\sqrt{4-\frac{2r}{R}+\frac{r^2(R-2r)}{4R^2(R-r)}}+Q \end{aligned} \] where \(S\) is the area of the triangle and \(Q=(a-b)^2+(b-c)^2+(c-a)^2\) are given. They are both equivalent to a third inequality which involves the semiperimeter \(s\), the circumradius \(R\) and the inradius \(r\) of the triangle. For the proof, the author uses technique from [\textit{M. Lukarevski} and \textit{D. S. Marinescu}, J. Math. Inequal. 13, No. 3, Article No. 13--57, 827--832 (2019; Zbl 1429.51006)] to introduce first the parameter \(t=r/R\) and then to use the fundamental triangle inequality [\textit{M. Lukarevski}, Math. Mag. 96, No. 2, 141--149 (2023; Zbl 07687517)] \[ s^2 \leq 2R^2+10Rr-r^2+2(R-2r)\sqrt{R^2-2Rr}. \]