Summary — Book III (Propositions 54–76 [III. 1–56])

1. Five Thematic Pillars of Book III

  1. Tangent & Diameter Area Equalities (`III.1–15 [Props. 54–56]`): Equal triangles cut off by two tangents and the diameters through their contact points.

  2. Power of a Point & Intersecting Chords (`III.16–29 [Props. 57–61]`): for chords in fixed directions—often called Newton’s Theorem despite originating verbatim in Apollonius Book III!

  3. Harmonic Poles & Polars (`III.30–40 [Props. 62–66]`): Every secant through pole T is divided harmonically by the conic and polar Qq: .

  4. Intercepts on Two Tangents by a Third (`III.41–44 [Props. 67–68]`): Projective homography intercepted on two fixed tangents by a variable third tangent ( on central conics; proportional division on the parabola—the exact Bezier/Steiner string construction!).

  5. Central Foci & the 3-/4-Line Locus (`III.45–56 [Props. 69–76]`): - Focal application ; equal reflection angles (III.48); pedal circle (III.49); (III.51–52). - Constant intercept products (III.53–56) synthesizing the Three-Line and Four-Line Locus.