Unit 4 Summary: Inversion, Casey’s Theorem, Feuerbach & Polar Reciprocity¶
Mathematical Synthesis of Casey’s Book VI, Sections IV & VII and Book III Props. 25, 27
1. Crown Jewels of Casey’s Sequel¶
Unit 4 contains the mathematical climax of Casey’s treatise:
Conformal Circle Inversion: $P \mapsto O + \frac{k^2}{|OP|^2}(P - O)$ linearizes circles through $O$, fixes orthogonal circles globally, preserves angles and cross-ratios, and leaves normalized bitangent quadrances $t_{12}^2 / (r_1 r_2)$ invariant.
Casey’s Generalized Ptolemy Theorem (1866): Four circles tangent to a fifth satisfy $t_{12} t_{34} \pm t_{14} t_{23} \pm t_{13} t_{24} = 0$, proved in six lines by inverting from a contact point on the fifth circle into Euler’s collinear four-point identity!
Feuerbach’s Theorem at $F_e = X(11)$: Proved both algebraically (as the point-circle midpoint specialization $\frac{1}{4}[a(c-b) + b(a-c) - c(a-b)] = 0$ of Casey’s Theorem) and synthetically (via Casey’s 1861 orthogonal circle $(M_a, \frac{1}{2}|b-c|)$ inverting the fourth common tangent $[C T_s]$ of $(I)$ and $(I_a)$ onto the Nine-Points Circle!).
Pole–Polar Bilinear Duality: $(x_0 - h)(x - h) + (y_0 - k)(y - k) = r^2$ is symmetric in $(x_0, y_0) \leftrightarrow (x, y)$, immediately establishing La Hire’s Reciprocity, self-conjugate triangles $\triangle DUV$, and polar reciprocation (Pascal $\leftrightarrow$ Brianchon).
2. Verified Exact Symbolic Ledgers (geometor.casey.feuerbach & geometor.casey.poles_polars)¶
Theorem / Element |
Symbol |
Exact Symbolic Value |
|---|---|---|
Incentre & Inradius on $(15, 13, 14)$ |
$I = X(1), \; r$ |
$I = (6, 4), \; r = 4$ |
Nine-Points Centre & Radius |
$N = X(5), \; R_9$ |
$N = \left(6, \frac{63}{16}\right), \; R_9 = \frac{65}{16}$ |
Internal Centre Distance |
$|N I| = R_9 - r$ |
$\frac{1}{16} = \frac{65}{16} - 4$ |
Feuerbach Point |
$F_e = X(11)$ |
$F_e = (6, 8)$ |
Three Excircle Feuerbach Touch Points |
$F_a, \; F_b, \; F_c$ |
$\left(\frac{2709}{289}, \frac{1792}{289}\right), \; \left(\frac{553}{233}, \frac{1344}{233}\right), \; \left(\frac{1672}{257}, -\frac{24}{257}\right)$ |
Casey Orthogonal Circle & Contact Inverse |
$(M_a, T_a), \; T_s \mapsto F_e$ |
Centre $\left(\frac{19}{2}, 6\right)$, radius $\frac{1}{2}$; $T_s = \left(\frac{614}{65}, \frac{392}{65}\right) \mapsto (6, 8)$ |
Base Circle & Pole–Polar Pair (0071.g1) |
$x^2 + y^2 = 25, \; D \leftrightarrow [UV]$ |
$D = \left(\frac{15}{2}, \frac{5}{4}\right), \; 6x + y - 20 = 0$ |
Self-Conjugate Triangle & Reciprocal Pole |
$\triangle DUV, \; Q \in [UV]$ |
$U = (3, 2), \; V = \left(\frac{5}{3}, 10\right), \; Q = (4, -4)$ |