Summary — Unit IV: Collinear Centers, 60° Family & Angle Trisection (§§ 25–34)¶
Analytical Synopsis of §§ 25–34 (pp. 117–123)
Isosceles Characterization of Euler-Line Collinearity (§§ 25–29): The incenter \(G = X(1)\) lies on the Euler Line \(EH\) if and only if \(\triangle ABC\) is isosceles (\(a = b\)). Setting \(EH = 2d\) around the midpoint of \(EH\) (namely the nine-point center \(L = X(5)\)!) and \(EG = d \mp e\):
When \(a > c\) (tall isosceles, Fig. 8), \(EG < d = \frac{1}{2}EH\), so \(G\) lies between \(E\) and the nine-point center!
When \(a < c\) (wide isosceles, Fig. 9), \(d < EG < \frac{4}{3}d = EF\) (\(d > 3e\)), so \(G\) lies strictly between the nine-point center and the centroid \(F\)!
Symmetric (e, f, k) = (EG, GH, EH) Parameterization (§§ 30–31): Eliminating the centroid \(F\) expresses \(R, Q, P\) directly in terms of the three sides \((e, f, k)\) of \(\triangle EGH\), requiring \(2e^2 + f^2 < k^2 < 2e^2 + 2f^2\) (equivalent to \(Q > 0\) and \(R > 0\), i.e., \(EG^2 < 2FH^2 + FG^2\) and \(GH^2 > 0\)).
Two Compass-Constructible vs. Trisection Regimes (§§ 32–34):
Equidistant Locus EG = GH (§§ 32–33): Whenever \(e = f\) (\(X(1)\) equidistant from \(X(4)\) and \(X(3)\)), one angle of \(\triangle ABC\) equals \(60^\circ\) (\(\cos\gamma = \frac{1}{2}\)), giving \(c = R_{\text{circ}}\sqrt{3}\) and splitting the cubic into a rational linear factor and a quadratic!
General Casus Irreducibilis (§ 34): For generic \((e, f, k)\) such as \((e^2, f^2, k^2) = (3, 2, 9)\), the cubic \(z^3 - \sqrt{71}\,z^2 + 22z - 2\sqrt{71} = 0\) is irreducible with three real positive roots, requiring trigonometric angle trisection.