Summary — Books VI & VII (Propositions 114–147 [VI. 1–33 & VII. 1–51])

1. Book VI Key Results (Props. 114–122)

  • Definitions 1–6: Exact definitions of equal/unequal and similar/dissimilar conic sections and conic segments.

  • Similarity Invariants (`VI.11–13 [Props. 115–116]`): - Every parabola is similar to every other parabola. - Two central conics of the same species are similar iff ().

  • 3D Cone Embedding (`VI.28–33 [Props. 120–122]`): Explicit straightedge-and-compass construction of the cutting planes in a given right circular cone that cut out a prescribed parabola, ellipse, or admissible hyperbola (diorismos on the hyperbola’s asymptotic angle vs. the cone’s apex angle!).

2. Book VII Key Invariants & Extrema (Props. 123–147)

  • Homothety Lemmas (`VII.1–11 [Props. 123–128]`): Auxiliary axis division translates oblique conjugate diameters into collinear segments on AA.

  • First Conjugate Diameter Theorem (`VII.12–13 [Prop. 129]`): .

  • Second Conjugate Diameter Theorem (`VII.31 [Prop. 136]`): ().

  • Diorismos Inequalities (`VII.21–51 [Props. 131–147]`): Exact extrema of , , , , and preparing the solvability conditions for reconstructing conics from conjugate-diameter data in the lost Book VIII.