Unit III · Executive Geometric Summary¶
Core Results of Liber Primus XXIII–XXVI & Liber Secundus (1678, pp. 36–72)¶
Seven-Line 3D Tetrahedral Equilibrium (*Lib. I, Prop. XXIII*): Given non-coplanar vertices $A, B, C, D \in \mathbb{Q}^3$ with non-zero masses $(m_A, m_B, m_C, m_D)$ and $M = m_A + m_B + m_C + m_D \neq 0$, the unique spatial center of gravity $O = \frac{1}{M}(m_A A + m_B B + m_C C + m_D D)$ lies simultaneously on all 7 static lines (
solve_tetrahedron_static_equilibrium()):4 Vertex-to-Face Cevians $V G_V$, divided at $O$ in ratio $\frac{\overline{VO}}{\overline{OG_V}} = \frac{M - m_V}{m_V}$;
3 Opposite-Edge Bimedians $E_{ij} E_{kl}$, divided at $O$ in ratio $\frac{\overline{E_{ij} O}}{\overline{O E_{kl}}} = \frac{m_k + m_l}{m_i + m_j}$.
Affine Invariance of Circumscribed Conic Tangency (*Lib. II, Lemma I*): Under any invertible affine transformation $X' = M X + b$ (realized geometrically by Serenus and Borelli via oblique cylindrical section), a circle maps to a general ellipse, tangent lines map to tangent lines, and every collinear contact division ratio $\lambda_k = \overline{V_k T_k} / \overline{T_k V_{k+1}}$ is strictly preserved, keeping $\prod_{k=0}^{n-1} \lambda_k = 1$ (
prove_circumscribed_polygon_lemma()).Gergonne $X(7)$ & Isotomic Conjugate $X(8)$ (*Lib. II, Prop. I & Corollarium*): Reflecting the contact feet $(D, E, F)$ of $P = (u : v : w)$ across the side midpoints produces $D^* = B + C - D$, $E^* = C + A - E$, $F^* = A + B - F$, inverting every side ratio and mapping $(u : v : w) \mapsto (u^{-1} : v^{-1} : w^{-1})$. For the incircle, $X(7) = \left(\frac{1}{s-a} : \frac{1}{s-b} : \frac{1}{s-c}\right) \mapsto X(8) = (s - a : s - b : s - c)$ (
construct_isotomic_conjugate()).Odd $(2m + 1)$-gon Multilateral Ceva Concurrency: On our convex rational pentagon $V_0=(0,0), V_1=(8,0), V_2=(11,6), V_3=(5,11), V_4=(-2,6)$ with $P=(4,4)$,
verify_odd_polygon_ceva()yields exact side ratios $\left(\frac{5}{6}, 2, \frac{13}{10}, \frac{3}{4}, \frac{8}{13}\right)$ whose product is identically $1$.