Critical & Symbolic Notes — Unit IV: Homothety & Derived Triangles (§§ 61–87)

Why Two Dual Pedal/Contact Operations Restore Parallel Sides

  1. Angle-Doubling & Homothety: In any acute triangle \(ABC\) with angles \((\alpha, \beta, \gamma)\):

    • Passing to the contact triangle \(\triangle DEF\) replaces \((\alpha, \beta, \gamma)\) by \(\left(90^\circ - \tfrac{1}{2}\alpha, 90^\circ - \tfrac{1}{2}\beta, 90^\circ - \tfrac{1}{2}\gamma\right)\).

    • Passing from \(\triangle DEF\) to its orthic triangle \(\triangle mnp\) (\(\mu\)) applies \(\theta \mapsto 180^\circ - 2\theta\), restoring \(180^\circ - 2(90^\circ - \tfrac{1}{2}\alpha) = \alpha\)! Moreover, the tangent-chord theorem [III. 32] aligns each side of \(\triangle mnp\) parallel to the corresponding side of \(\triangle ABC\) with homothety ratio \(\frac{r}{2R}\) (hence area ratio \(\frac{r^2}{4R^2}\), proving § 63 and § 65 simultaneously!).

  2. Exact Verification of § 64 on the Canonical (15, 13, 14) Triangle: With \(\Delta = 84\), \(R = \frac{65}{8}\), \(4R^2 = \frac{4225}{16}\), \((r, r_a, r_b, r_c) = \left(4, 14, \frac{21}{2}, 12\right)\), and \(\rho = \frac{99}{52}\):

    • \(\mu + \mu' + \mu'' + \mu''' = \frac{84}{4225/16}\left(16 + 196 + \frac{441}{4} + 144\right) = \frac{1344}{4225} \cdot \frac{1865}{4} = \frac{336 \cdot 373}{845} = \frac{125328}{845}\).

    • \(\triangle MNP = \frac{\rho}{R}\Delta = \frac{99/52}{65/8} \cdot 84 = \frac{198}{845} \cdot 84 = \frac{16632}{845}\), whence \(\sum \mu^{(k)} + \triangle MNP = \frac{125328 + 16632}{845} = \frac{141960}{845} = 168 = 2\Delta\).