Disuguaglianze isoperimetriche per i triangoli

Baiocchi, Claudio

Rendiconto della Accademia delle scienze fisiche e matematiche, Série 4, Tome 85 (2018) no. 1, pp. 173-176

Résumé

The well known isoperimetric inequality says that for any triangle $ABC$ the semiperimeter $p$ and the surface $S$ satisfy: \begin{equation*} p^{2} \geq 3\sqrt{3}*S \end{equation*} with equality if and only if $ABC$ is equilateral; in other words, among all triangles with prescribed area, the equilateral one has perimeter minimum. We want to get a more precise inequality, concerning the family of triangles having a prescribed angle $\alpha$; we will prove: \begin{equation*} p^{2} \geq \frac{2(1 + \sin \frac{\alpha}{2})^{2}}{\sin \alpha} * S \end{equation*} with equality if and only $\alpha$ is the angle opposite to the base of an isosceles triangle.