Summary — Book II (Propositions 28–53 [II. 1–53])

1. Core Theorems of Book II

• Existence & Commonality of Asymptotes (`II.1, 15, 17, 21 [Prop. 28]`): Half-tangent squared defines the two rays CLCL shared by both opposite branches and both conjugate hyperbolas.

• Tangent Bisection (`II.3, 9 [Prop. 30]`): Every tangent segment intercepted by the asymptotes is bisected at the point of tangency ().

• Secant Intercept Equality (`II.8, 10, 16 [Prop. 33]`): Every secant cuts equal segments between hyperbola and asymptotes () with constant product .

• Asymptotic Hyperbola Symptom (`II.12 [Prop. 34]`): ( along the asymptotes as oblique axes!).

• Asymptotic Convergence (`II.14 [Prop. 35]`): Distance from curve to asymptote decreases monotonically below any assigned magnitude.

• Conjugate Hyperbolas (`II.17–23 [Props. 36–39]`): Dual pairing of and , where conjugate diameters of one are conjugate diameters of the other.

• Constructive Problems (`II.44–53 [Props. 47–53]`): Euclidean straightedge-and-compass algorithms to recover diameters, center, principal axes, external tangents, and tangents making a given angle with the axis or diameter.