Unit II · Executive Geometric Summary

Core Results of Liber Primus, Props. VI–XXII (1678, pp. 7–35)

  1. Equivalence of Static & Synthetic Proofs (*Props.* VI–VII): Pietro Paolo Caravaggio Jr.'s parallel-line construction through vertex $A$ (Prop. VI) translates Ceva’s static weight cancellation into three pairs of similar triangles ([VI.4]), establishing both necessity and sufficiency (reductio / coincidence of the third foot $F^* = F$ by [V.9]).

  2. Ceva–Menelaus Duality & Straightedge Harmonic Conjugation (*Prop.* VIII): Pairing concurrent cevians $AD, BE, CF$ ($\frac{AF}{FB}\frac{BD}{DC}\frac{CE}{EA} = +1$) with the transversal $EF$ cutting $BC$ at $D'$ ($\frac{AF}{FB}\frac{BD'}{D'C}\frac{CE}{EA} = -1$) yields $\frac{BD}{DC} + \frac{BD'}{D'C} = 0$, proving $(B, C; D, D') = -1$ (construct_cevian_menelaus_harmonic()).

  3. Exact Rational Constellation on Heronian $\triangle(13, 14, 15)$: Choosing $A=(5,12), B=(0,0), C=(14,0)$ ($a=14, b=15, c=13, s=21, \Delta=84, r=4$) places every classical center inside $\mathbb{Q}^2$ with zero radicals:

Canonical Triangle Centers on $\triangle(13, 14, 15)$ (geometor.ceva.static_barycentrics)

Center

Geometric Definition

Static Weights $u : v : w$

Exact Coordinates in $\mathbb{Q}^2$

Ceva Product

X(1)

Incenter (angle bisectors)

$14 : 15 : 13$

$(6, 4)$

$\frac{15}{14}\cdot\frac{13}{15}\cdot\frac{14}{13} = 1$

X(2)

Centroid (medians)

$1 : 1 : 1$

$\left(\frac{19}{3}, 4\right)$

$1 \cdot 1 \cdot 1 = 1$

X(4)

Orthocenter (altitudes)

$70 : 99 : 55$

$\left(5, \frac{15}{4}\right)$

$\frac{99}{70}\cdot\frac{55}{99}\cdot\frac{70}{55} = 1$

X(6)

Symmedian point ($a^2:b^2:c^2$)

$196 : 225 : 169$

$\left(\frac{1673}{295}, \frac{1176}{295}\right)$

$\frac{225}{196}\cdot\frac{169}{225}\cdot\frac{196}{169} = 1$

X(7)

Gergonne point (Lib. II Prop. I)

$24 : 28 : 21$

$\left(\frac{414}{73}, \frac{288}{73}\right)$

$\frac{7}{6}\cdot\frac{3}{4}\cdot\frac{8}{7} = 1$

X(8)

Nagel point (isotomic)

$7 : 6 : 8$

$(7, 4)$

$\frac{6}{7}\cdot\frac{8}{6}\cdot\frac{7}{8} = 1$

J_A

$A$-Excenter (Elem. III F - A)

$-14 : 15 : 13$

$(8, -12)$

$\left(-\frac{15}{14}\right)\left(\frac{13}{15}\right)\left(-\frac{14}{13}\right) = 1$

  1. Spread Ceva’s Theorem: On X(1), verify_spread_ceva() computes the six exact vertex spreads over $\mathbb{Q}$ ($s(\angle BAD) = s(\angle DAC) = \frac{16}{65}$, $s(\angle CBE) = s(\angle EBA) = \frac{4}{13}$, $s(\angle ACF) = s(\angle FCB) = \frac{1}{5}$), confirming unit spread product and unit quadrance product with zero square roots.