Symbolic substitution has a geometric meaning (Q2878858)

Let \(\Delta=\{A,B,C\}\) be a triangle of \({\mathbb R}^2\) having in the canonical Euclidean metric the side lengths \(a,b,c\). The author endows \({\mathbb R}^2\) with a generalized metric (= \(g\)-metric) \(d_g\) by prescribing the \(d_g\)-lengths of the sides of \(\Delta\) as the positive real numbers \(a_g,b_g,c_g\). Depending on NEWLINE\[NEWLINE \sigma_g:=(a_g+b_g+c_g)(-a_g+b_g+c_g)(a_g-b_g+c_g)(a_g+b_g-c_g) NEWLINE\]NEWLINE the author gets for \(\sigma_g>0\) an affine version of the Euclidean, for \(\sigma_g=0\) a Galilean, and for \(\sigma_g<0\) a Lorentz-Minkowski metric. If \(x(a,b,c)\) is a barycentric center function of a triangle center \(X\), then the corresponding \(g\)-center \(X_g\) has the following homogenous coordinates with respect to \(\Delta\): NEWLINE\[NEWLINE X_g=\bigl(x(a_g,b_g,c_g):x(b_g,c_g,a_g):x(c_g,a_g,b_g)\bigr). NEWLINE\]NEWLINE For each of the three cases \(g\)-circumcenter, \(g\)-orthogonality, and \(g\)-angles are seized by formulas. ''By comparing two different metrics in the affine plane, it is shown that the symbolic substitution by \textit{C. Kimberling} [Aequationes Math. 73, No. 1--2, 156--171 (2007; Zbl 1119.39022)] has a clear geometric meaning.'' Finally, the author presents generalized versions of Feuerbach's conic theorem.