GC-04.02 · Johannes Kepler’s Geometric-Progression Right Triangle (1597)

Discoverer(s): Johannes Kepler | Date: 1597 | Field: \(\mathbb{Q}(\sqrt{\phi})\) | Loci: 2 (Lemoine \(S = 6\))

Divide hypotenuse AB at H in the golden ratio (\(|AH|/|HB| = \phi\)), and erect perpendicular HC to semicircle cOB on diameter AB. Right triangle ABC has side lengths in geometric progression \(1 : \sqrt{\phi} : \phi\) and side quadrances in progression \(1 : \phi : \phi^2\).
GC-04.02 — Johannes Kepler's Geometric-Progression Right Triangle (1597) cOB: y**2 + (x - 3/4 - sqrt(5)/4)**2 - (sqrt(5)/4 + 3/4)**2 = 0 AB: y = 0 Golden major AH = 1/2 + sqrt(5)/2 Golden minor HB = 1 O = (sqrt(5)/4 + 3/4, 0) C = (1/2 + sqrt(5)/2, sqrt(1/2 + sqrt(5)/2)) A = (0, 0) H = (1/2 + sqrt(5)/2, 0) B = (sqrt(5)/2 + 3/2, 0) O C A H B

Given & Constructive Steps

Given: Golden-sectioned hypotenuse A \(= (0,0)\), H \(= (\phi, 0)\), B \(= (\phi^2, 0)\) with midpoint O \(= (\tfrac{\phi^2}{2}, 0)\).

step Draw circle cOB on diameter AB of length \(\phi^2 = \phi + 1\).
step Erect altitude HC perpendicular to AB at golden division point H to meet cOB at C \(= (\phi, \sqrt{\phi})\).
step Join AC and BC to complete right triangle ABC.

Exact Symbolic Proof & Certificate

By Euclid VI.8 and VI.13, altitude HC is the geometric mean \(|HC|^2 = |AH|\cdot|HB| = \phi \cdot 1 = \phi\). Thus \(Q(B, C) = 1 + \phi = \phi^2\), \(Q(A, C) = \phi^2 + \phi = \phi^3\), and \(Q(A, B) = \phi^4\). Because \(\phi^2 + \phi^3 = \phi^2(1 + \phi) = \phi^4\), Kepler’s two ‘treasures of geometry’—the Pythagorean theorem and the golden section—unite uniquely in \(\triangle ABC\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

AH \(/\) HB \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (A, H, B)

Exact Segment Lengths

\(|AH| = \frac{1}{2} + \frac{\sqrt{5}}{2}\), \(|HB| = 1\)

Rational Trigonometry Quadrances

\(Q(A, H) = \frac{\sqrt{5}}{2} + \frac{3}{2}\), \(Q(H, B) = 1\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

\(\mathbb{Q}(\sqrt{\phi})\)

Géométrographie Loci

2 given points · 1 lines · 1 circles · 1 polygons (elementary loci \(= 2\), Lemoine \(S = 6\))

Automated Divine Scan

geometor.divine.golden detected 1 exact collinear golden section(s) and 0 connected golden chain(s) in this model:

  • Section (A, H, B): long \(|AH|^2 = \frac{\sqrt{5}}{2} + \frac{3}{2}\) vs. short \(|HB|^2 = 1\)

Historical Commentary & Attribution

Kepler’s triangle is the canonical entry point in the catalog whose coordinates genuinely require the quadratic extension \(\mathbb{Q}(\sqrt{\phi})\) of degree \(4\) over \(\mathbb{Q}\).