Unit 2 Summary: Power of a Point, Orthogonal Circles & Coaxal Pencils¶
Mathematical Synthesis of Casey’s Book III (Props. 20–24, 28) & Book VI, Section V
1. Algebraic & Synthetic Invariants¶
Unit 2 develops the quadratic algebra of circles over the Euclidean plane:
Power of a Point: $\operatorname{pow}(P, C) = (x_P - h)^2 + (y_P - k)^2 - r^2$. Positive outside $C$ ($= |PT|^2$), zero on $C$, negative inside $C$ ($= -PA \cdot PB$).
Orthogonality Criterion: $C_1 \perp C_2 \iff |O_1 O_2|^2 = r_1^2 + r_2^2 \iff \operatorname{pow}(O_1, C_2) = r_1^2$.
Radical Axis as Linear Difference: Subtracting $C_2(x, y) = 0$ from $C_1(x, y) = 0$ eliminates $x^2 + y^2$, yielding the linear radical axis $2(h_2 - h_1)x + 2(k_2 - k_1)y + (h_1^2 + k_1^2 - r_1^2) - (h_2^2 + k_2^2 - r_2^2) = 0$.
Casey’s Difference-of-Powers Identity (III.24): Choosing coordinates with line of centres along the $x$-axis and radical axis at $x = 0$, every circle in the coaxal pencil takes the canonical form
$$ x^2 + y^2 - 2\lambda x + \delta = 0, $$
where $\operatorname{pow}(P, C_\lambda) - \operatorname{pow}(P, C_\mu) = 2(\mu - \lambda)x_P = 2\,|O_\mu O_\lambda|\cdot PN$. When $\delta = \ell^2 > 0$, the pencil is non-intersecting with real Poncelet Limiting Points $L_1 = (-\ell, 0)$ and $L_2 = (+\ell, 0)$ at $\lambda = \mp\ell$, and every circle $x^2 + y^2 - 2\eta y - \ell^2 = 0$ centered on $x = 0$ passes through $L_1, L_2$ and cuts every $C_\lambda$ orthogonally!
2. Verified Exact Construction Ledger (geometor.casey.coaxial)¶
Constructed in construct_radical_axis_and_coaxial_pencil() (0069.g1.svg):
Geometric Element |
Symbol / Equation |
Exact Rational Coordinates |
|---|---|---|
Given Non-Intersecting Circles |
$C_1 = (O_1, 6), \; C_2 = (O_2, 15)$ |
$O_1 = (-10, 0), \; O_2 = (17, 0)$ ($\delta = 64$) |
Third Pencil Member |
$C_3 = (O_3, 6)$ |
$O_3 = (10, 0)$ ($10^2 - 6^2 = 64$) |
Auxiliary Secant Circle |
$C_{\text{aux}} = ((2, 8), 10)$ |
Meets $C_1, C_2$ in rational chords concurring at $R_c$ |
Radical Centre & Radical Axis |
$R_c, \; [D R_c]$ |
$R_c = (0, -6), \; x = 0, \; D = (0, 0)$ |
Tangent Contact on $C_1$ from $D$ |
$T_1, \; H$ |
$T_1 = \left(-\frac{32}{5}, \frac{24}{5}\right), \; H = \left(-\frac{32}{5}, 0\right), \; |DT_1| = 8$ |
Poncelet Limiting Points |
$L_1, \; L_2$ |
$L_1 = (-8, 0), \; L_2 = (8, 0)$ |
Conjugate Orthogonal Circle |
$(R_c, L_1)$ |
Centre $(0, -6)$, radius $10$ ($C_{\perp} \perp C_1, C_2, C_3$) |