Summary — Book V (Propositions 84–113 [V. 1–77])

1. Subnormals, Evolute Equations & Normal Multiplicity

Conic	Subnormal NG	Evolute Equation Derived from Prop. 99 [V. 51–52]	Real Normals from O
Parabola ()	(constant)	(Semicubical Parabola)	3 (inside), 2 (on cusp-arc), 1 (outside)
Ellipse ()		(Astroidal Lamé Curve)	4 (inside), 3 (on arc), 2 (outside / at cusps)
Hyperbola ()			4, 3, or 2 across branches

2. The Hyperbola of Apollonius (V. 62–63 [Prop. 106])

For an ellipse and target point Ox0y0, the normal at Pxy passes through O if and only if P lies on the auxiliary rectangular hyperbola:

\[(a^2 - b^2)\, x y + b^2 y_0\, x - a^2 x_0\, y = 0,\]

which passes through the center C00, the target point Ox0y0, and has asymptotes parallel to the principal axes—intersecting the conic in at most 4 points by Book IV Prop. 78!