GC-04.02 · Symbolic Model & Source

Companion symbolic Python construction for GC-04.02 · Johannes Kepler’s Geometric-Progression Right Triangle (1597) (GC-04.02).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-04.02")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_kepler_geometric_triangle() -> ConstructionResult:
            """Johannes Kepler's right triangle with sides in geometric progression (1597)."""
            m = new_catalog_model("kepler-geometric-triangle")
            phi = sp.Rational(1, 2) + sp.sqrt(5) / 2
            phi2 = phi + 1

            A = m.set_point(0, 0, classes=["given"], ID="A")
            H = m.set_point(phi, 0, classes=["golden"], ID="H")
            B = m.set_point(phi2, 0, classes=["given"], ID="B")
            O = m.set_point(phi2 / 2, 0, ID="O")

            ln_ab = m.construct_line(A, B, classes=["golden"], ID="AB")
            _ = ln_ab
            m.construct_circle(O, B, classes=["primary"], ID="cOB")

            # Geometric mean altitude |HC| = sqrt(|AH| * |HB|) = sqrt(phi)
            C = m.set_point(phi, sp.sqrt(phi), classes=["golden"], ID="C")
            m.construct_polygon([A, B, C], classes=["golden"], ID="ABC")
            m.construct_segment(H, C, classes=["auxiliary"], ID="HC")

            return make_result(m, ("A", "H"), ("H", "B"), ("A", "H", "B"))
        

Exact Symbolic Point Coordinates

All points in kepler-geometric-triangle are certified free of floating-point approximations over \(\mathbb{Q}(\sqrt{\phi})\):

Point	Exact \(x\)	Exact \(y\)
A	\(0\)	\(0\)
H	\(\frac{1}{2} + \frac{\sqrt{5}}{2}\)	\(0\)
B	\(\frac{\sqrt{5}}{2} + \frac{3}{2}\)	\(0\)
O	\(\frac{\sqrt{5}}{4} + \frac{3}{4}\)	\(0\)
C	\(\frac{1}{2} + \frac{\sqrt{5}}{2}\)	\(\frac{\sqrt{2} \sqrt{1 + \sqrt{5}}}{2}\)

Certified Golden Ratio: AH / HB = GoldenRatio.