GC-04.01 · Symbolic Model & Source

Companion symbolic Python construction for GC-04.01 · H. E. Huntley’s Golden Ratio in the 3-4-5 Right Triangle (1970) (GC-04.01).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-04.01")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_huntley_egyptian_345() -> ConstructionResult:
            """H. E. Huntley's unexpected appearance of phi in a 3-4-5 right triangle."""
            m = new_catalog_model("huntley-egyptian-345")
            C = m.set_point(0, 0, classes=["given"], ID="C")
            B = m.set_point(3, 0, classes=["given"], ID="B")
            A = m.set_point(0, 4, ID="A")

            m.construct_polygon([A, B, C], ID="ABC")

            # Angle bisector of <ABC meets AC at O=(0, 3/2); circle O(C) is tangent to AB at T=(6/5, 12/5)
            O = m.set_point(0, sp.Rational(3, 2), ID="O")
            T = m.set_point(sp.Rational(6, 5), sp.Rational(12, 5), ID="T")
            _ = T

            ln_bo = m.construct_line(B, O, classes=["golden"], ID="BO")
            c_oc = m.construct_circle(O, C, classes=["golden"], ID="cOC")

            m.intersect(c_oc, ln_bo, between=(B, O), classes=["golden"], ID="P")
            m.intersect(c_oc, ln_bo, beyond=(B, O), classes=["golden"], ID="Q")

            return make_result(m, ("P", "Q"), ("B", "P"), ("B", "P", "Q"))
        

Exact Symbolic Point Coordinates

All points in huntley-egyptian-345 are certified free of floating-point approximations over \(\mathbb{Q}(\phi)\):

Point	Exact \(x\)	Exact \(y\)
C	\(0\)	\(0\)
B	\(3\)	\(0\)
A	\(0\)	\(4\)
O	\(0\)	\(\frac{3}{2}\)
T	\(\frac{6}{5}\)	\(\frac{12}{5}\)
P	\(\frac{3 \sqrt{5}}{5}\)	\(\frac{3}{2} - \frac{3 \sqrt{5}}{10}\)
Q	\(- \frac{3 \sqrt{5}}{5}\)	\(\frac{3 \sqrt{5}}{10} + \frac{3}{2}\)

Certified Golden Ratio: PQ / BP = GoldenRatio.