Encounter between classical mean values and conics (Q2036216)

The geometric representation of the classical means \(M(p,q)\) (arithmetic, geometric and harmonic) of two numbers \(p\) and \(q\) on a circle is known since classical antiquity. In this nicely written article, the author considers generalisations of the representations of the means to arbitrary conic sections. Let \(c\) be a conic section, \(S\) a point not on \(c\), and \(s\) a line through \(S\) which intersects \(c\) in \(P\) and \(Q\). Let \(M(P,Q)\) be the point on \(s\) between \(P\) and \(Q\) such that its distance from \(S\) is \(M(p,q)\) where \(p=|SP|\), \(q=|SQ|\). The locus of the point \(M(P,Q)\) is described when the line \(s\) rotates around the point \(S\). That is accomplished using different methods: geometric (affine and projective) and algebraic. These approaches are then compared in the conclusion. The article contains carefully drawn diagrams which make the reading enjoyable.