F.31 Logic Tests

Boolean Distance Inequality Triples and Signed Collinear Division Ratios

Given collinear points A, B, C, the three Boolean inequality tests T_1 = (|CA| < |AB|), T_2 = (|CB| < |AB|), and T_3 = (|CA| < |CB|) uniquely partition the line AB into Region A (-1 < k < 0), interior segment AB (0 < k < infty), and Region B (-infty < k < -1), determining the sign of k = +/- |AC| / |BC|.
ABC c a b / A P_{in} / / A P_{out} / / B P_{in} / / B P_{out} / / C P_{in} / / C P_{out} / A (0.000, 0.000) B (7.000, 0.000) C (4.286, 4.199) P_{in} (3.837, 1.800) P_{out} (6.714, 3.149) AA BB CC PinPin PoutPout + − ⟲ Lbl Gde

Interactive GEOMETOR construction for F.31 (Logic Tests). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== l = line(A, B) s_ab = segment(A, B) s_ac = segment(A, C) s_bc = segment(B, C)

Mathematical Synthesis

When automating harmonic conjugate and Apollonian circles calculations in C.a.R. and OpenDocument spreadsheets (SeeligNotebooks), Euclidean distance functions return unsigned magnitudes \(\|CA\|, \|CB\|, \|AB\|\). In this six-page treatise (Three Logical Tests — Boolean outputs 1=True / 0=False, numbered 1/–6/), Seelig defines three elementary predicates \(T_1: \|CA\| < \|AB\|\), \(T_2: \|CB\| < \|AB\|\), and \(T_3: \|CA\| < \|CB\|\).

Enumerating all \(2^3 = 8\) truth assignments across Scenarios I–VI, Seelig shows that Region A (external on the A-side, \(-1 < k < 0\)) is characterized by \(\overline{T}_2 T_3\), the interior of segment A B (\(0 < k < \infty\)) by \(T_1 T_2\), and Region B (external on the B-side, \(-\infty < k < -1\)) by \(\overline{T}_1 \overline{T}_3\), compiling nested if(...) formulas for both C.a.R. (Folio 4) and Excel/Calc (Folios 5–6).

Key Algebraic Identities

\[T_1 = (\|CA\| < \|AB\|), \qquad T_2 = (\|CB\| < \|AB\|), \qquad T_3 = (\|CA\| < \|CB\|)\]
\[\text{Region } A:\; \overline{T}_2 T_3\; (-1 < k < 0), \qquad \text{Region } AB:\; T_1 T_2\; (0 < k < \infty), \qquad \text{Region } B:\; \overline{T}_1\overline{T}_3\; (-\infty < k < -1)\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios

F.31 Logic Tests — Folios (Folios 0215–0220, 6 scanned manuscript pages in resources/seelig/31_logic-tests)

Euclid’s Elements

V.8 (EuclidElements)

GEOMETOR Elements (G-Index)

G.5 surface

Encyclopedia of Triangle Centers

General projective / metric locus (Kimberling1998)

Seelig Spreadsheets

Point in Triangle, Point in Triangle (2), Harmonic Conjugates (Spreadsheets)

Constructive Dependency Graph

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Prerequisites: F.2 Bilinear Transformations, F.25 Harmonic Conjugates