The elementary geometric potential of the Euler inequality (Q524956)

This is a very informative look at the many possible proofs of Euler's inequality (\(R\geq 2r\)), both purely geometric and algebraic, depending or not upon Euler's formula \(OI^2=R(R-2r)\) (itself proved in many different ways, one of which is based on inversion), as well as its generalizations (such as the Erdős-Mordell theorem), extensions to \(n\)-gons, and analogues in three-dimensional space. On the historical side, the author mentions only the problem of the originator of the inequality, given that Euler's paper usually mentioned in this connection does not contain the inequality in its current form, but rather only \(OI^2=\frac{(abc)^2}{16S^2}-\frac{abc}{a+b+c}\) (where \(S\) denotes the area of the triangle). There is no mention of the first authors of the many different proofs of the inequality that are presented in this paper.