Unit III · 3D Tetrahedra, Circumscribed Conics, Isotomic Conjugates & Odd Polygons (pp. 36–72)

Original GEOMETOR English translation from Giovanni Ceva’s De lineis rectis se invicem secantibus statica constructio (Milan: Ludovico Monza, 1678, Liber Primus, Props. XXIII–XXVI, pp. 36–43, & Liber Secundus, Lemma I & Props. I–XIV, pp. 45–72; Tab. 3–7, Figs. 28–61).

1. Liber Primus, Props. XXIII–XXVI — The Seven-Line Tetrahedron Equilibrium (pp. 36–43)

Crowning Liber Primus (Tab. 3, Figs. 28–31), Giovanni Ceva ascends from the plane into three-dimensional solid geometry. Suppose four arbitrary non-zero weights $(m_A, m_B, m_C, m_D)$ are placed at the four vertices A, B, C, D of a solid tetrahedron ABCD. By associative static grouping, the global spatial center of gravity O can be reached in seven distinct ways:

  1. Four Spatial Vertex-to-Face Cevians: First combine any three vertices on their triangular face—yielding the four static face centers of gravity GA on $\triangle BCD$, GB on $\triangle ACD$, GC on $\triangle ABD$, and GD on $\triangle ABC$—and then join each face center to the remaining opposite vertex (AGA, BGB, CGC, DGD). Every one of these four spatial rays divides at O in exact ratio $\overline{VO} : \overline{OG_V} = (m_{\text{tot}} - m_V) : m_V$ (for equal weights $1:1:1:1$, the classic $3 : 1$ tetrahedral centroid ratio of Commandino, 1565).

  2. Three Skew-Edge Bimedians: Alternatively, pair the four vertices into two opposite skew edges—$(AB, CD)$, $(AC, BD)$, or $(AD, BC)$—balancing each edge at its lever fulcrum (EAB with ECD, EAC with EBD, and EAD with EBC). Then the three straight segments (BABCD, BACBD, BADBC) joining opposite skew-edge fulcrums also pass through the exact same point O!

A (m=2) B (m=3) C (m=5) D (m=8) O Liber Primus Prop. XXIII (Tab. 3, Fig. 28): 3D Tetrahedron 7-Line Equilibrium ('mira quadam concordia') 4 Face-Barycenter Cevians (AG_A..DG_D) + 3 Skew-Edge Bimedians Concur at Unique Center O

Marveling at this simultaneous spatial intersection of seven lines in three-space, Ceva exclaims that all seven rays meet at O “mira quadam concordia”—with a wondrous concord! In Propositions XXIV–XXVI, Ceva extends the same barycentric decomposition to quadrangular and polygonal pyramids.

2. Liber Secundus, Lemma I — Borelli–Serenus Cylindrical Ellipse Projection (pp. 45–47)

Opening Liber Secundus (Tab. 4, Figs. 32–35), Ceva proves a master projective-affine lemma for polygons circumscribed about circles and ellipses:

  • Circle Case: Let any polygon of $n$ sides (whether $n$ be odd or even) be circumscribed around a circle circleC. Since the two tangents drawn from each exterior vertex $V_k$ to the circle have equal lengths $t_k$ ([III.36]), the contact points $T_0, T_1, \dots, T_{n-1}$ divide the consecutive sides into segments $(t_0, t_1), (t_1, t_2), \dots, (t_{n-1}, t_0)$ whose cyclic compound ratio is identically $1$:

    $$\frac{t_0}{t_1} \cdot \frac{t_1}{t_2} \cdots \frac{t_{n-1}}{t_0} = 1.$$

  • Ellipse Case via Cylindrical Section: Now erect a right or oblique circular cylinder upon circleC and cut it by an oblique plane (projCyl), producing an arbitrary Apollonian ellipse ellipseE, as demonstrated by Serenus of Antinoöpolis (1566) and Giovanni Alfonso Borelli (1661). Because parallel generator lines along the cylinder map every tangent side $V_k V_{k+1}$ to a tangent side $V'_k V'_{k+1}$ of the ellipse and preserve the ratio of collinear segments ([VI.2]), any $n$-gon circumscribed about an ellipse also has its sides divided at the contact points in cyclic compound ratio $1$ (sectio cylindrica)!

3. Liber Secundus, Prop. I & Corollarium — Contact Point $X(7)$ & Isotomic Conjugate $X(8)$

Applying Lemma I to a triangle ABC circumscribed around an inscribed circle incircle (or any inscribed ellipse), Proposition I (pp. 47–49, Tab. 4, Figs. 36–37) establishes immediately by Liber Primus, Prop. VII that the three cevians AD, BE, CF drawn to the points of contact concur at a single point X(7)—published by Ceva in 1678, 140 years before Joseph Diez Gergonne (punctum contactus)!

Immediately afterward, in Proposition I *Corollarium*, Ceva discovers isotomic conjugation (coniugatum isotomicum): if on each side of triangle ABC we transfer the contact distances to the opposite ends—choosing D*, E*, F* so that $BD^* = DC$ (BDeq), $CE^* = EA$ (CEeq), and $AF^* = FB$ (AFeq)—then each side ratio is inverted to its reciprocal, so their compound product remains $1^{-1} = 1$, and the three reflected cevians AD*, BE*, CF* concur at the reciprocal center X(8) (rediscovered by Christian Heinrich von Nagel in 1836)!

A (5,12) B (0,0) C (14,0) D D* E E* F F* X(7) X(8) Liber Secundus Prop. I & Corollarium: Contact Point X(7) (24:28:21) <-> Isotomic Nagel X(8) (7:6:8) Midpoint Reflection Swaps Ratios (AF/FB -> FB/AF), Preserving Unit Ceva Product = 1

4. Liber Secundus, Props. II–XIV — Multilateral Odd $(2m + 1)$-gon Concurrency (pp. 50–72)

In Propositions II–XIV (Tab. 4–7, Figs. 38–61), Ceva generalizes cevian concurrency and conic tangency from triangles to multilateral polygons of odd side count $n = 2m + 1$ (pentagons pentagon, heptagons, enneagons). Whenever rays (cev0, cev1, cev2, cev3, cev4) from the $2m + 1$ vertices (V0..:g:V4) pass through a common concurrency center P to meet their opposite sides at (Q0..:g:Q4), static weight transfer across the vertex-to-opposite-side bipartite cycle proves that the cyclic product of the $2m + 1$ opposite-side division ratios equals $+1$:

$$\prod_{i=0}^{2m} \frac{\overline{V_{i+m} Q_i}}{\overline{Q_i V_{i+m+1}}} = 1.$$

V_0 Q_0 V_1 Q_1 V_2 Q_2 V_3 Q_3 V_4 Q_4 P Liber Secundus: Odd (2m+1)-gon Ceva Concurrency Cyclic Opposite-Side Product = (5/6)*(2)*(13/10)*(3/4)*(8/13) = 1 Lib. II Lemma I: Borelli-Serenus Cylindrical Section Tab. 4, Figs. 32-35 — Affine Projection Preserves Contact Ratios = 1