Critical & Symbolic Notes — Unit I: The Four Tritangent Centers (§§ 1–18)

Feuerbach’s Sign-Flip Automorphism \(a \mapsto -a\) and Circumcircle Duality

1. The Algebraic Sign-Flip Principle (§ 8, p. 6): Already on page 6, Feuerbach explicitly articulates the algebraic transformation that governs his entire monograph: replacing \(+a\) by \(-a\) while holding \(+b, +c\) and the positive magnitude of \(\Delta\) fixed sends

   \[r = \frac{2\Delta}{a+b+c} \longmapsto r' = \frac{2\Delta}{-a+b+c}, \qquad R = \frac{abc}{4\Delta} \longmapsto \frac{(-a)bc}{4\Delta} = -R.\]

   Notice how carefully Feuerbach records +R zu -R on p. 6! In §§ 8, 11, 13, and 57 this rule works without flaw because \(R\) appears linearly (or after \(\rho\) has cancelled). In § 51 (p. 35), however, applying the same rule mechanically to \(\overline{OS}^2 = 2r^2 - 2\rho R\) without tracking that \(\rho\) also negates under \(a \mapsto -a\) produces the celebrated sign slip analyzed in Unit III Notes.

2. Circumcircle of ABC as Nine-Point Circle of the Excentral Triangle: In § 1 (p. 2) Feuerbach proves that \(\triangle ABC\) is the orthic triangle of the excentral triangle \(\triangle S'S''S'''\), with \(S = X(1)\) as its orthocenter. Combined with § 17 (\(SS'^2 + S''S'''^2 = (4R)^2\)), this means that the circumcircle \((ABC)\) of radius \(R\) is simultaneously the nine-point circle of \(\triangle S'S''S'''\), whose circumradius is \(R_{\text{ex}} = 2R\)!

3. Exact Benchmark Values on the Canonical (15, 13, 14) Triangle: In construct_fundamental_centers() and construct_feuerbach_tangency(), with \((a, b, c) = (15, 13, 14)\), \(\Delta = 84\), \(s = 21\), \(R = \frac{65}{8}\):

   • Tritangent radii: \(r = 4\), \(r' = r_a = 14\), \(r'' = r_b = \frac{21}{2}\), \(r''' = r_c = 12\).

   • Verification of § 2: \(rr'r''r''' = 4 \cdot 14 \cdot \frac{21}{2} \cdot 12 = 7056 = 84^2 = \Delta^2\).

   • Verification of § 5: \(r' + r'' + r''' - r = 14 + \frac{21}{2} + 12 - 4 = \frac{65}{2} = 4 \cdot \frac{65}{8} = 4R\).