GC-07.03 · Symbolic Model & Source

Companion symbolic Python construction for GC-07.03 · Jingcheng Tong’s Area-Bisecting Chord in a 1:3 Trapezoid (GC-07.03).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-07.03")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_tong_trapezoid_rms() -> ConstructionResult:
            """Jingcheng Tong's area-bisecting chord in a trapezoid of base ratio 1:3."""
            m = new_catalog_model("tong-trapezoid-rms")
            s5 = sp.sqrt(5)

            A = m.set_point(-1, 2, classes=["given"], ID="A")
            D = m.set_point(1, 2, ID="D")
            B = m.set_point(-3, 0, classes=["given"], ID="B")
            C = m.set_point(3, 0, ID="C")

            m.construct_polygon([A, B, C, D], ID="ABCD")
            ln_ab = m.construct_line(A, B, classes=["golden"], ID="AB")
            ln_cd = m.construct_line(D, C, classes=["auxiliary"], ID="DC")

            # Area-bisecting parallel chord MN of RMS length sqrt((2^2 + 6^2)/2) = 2*sqrt(5) at y = 3 - sqrt(5)
            M = m.set_point(-s5, 3 - s5, classes=["golden"], ID="M")
            N = m.set_point(s5, 3 - s5, classes=["golden"], ID="N")
            _ = (ln_ab, ln_cd)
            m.construct_line(M, N, classes=["golden"], ID="MN")

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

Exact Symbolic Point Coordinates

All points in tong-trapezoid-rms are certified free of floating-point approximations over \(\mathbb{Q}(\phi)\):

Point	Exact \(x\)	Exact \(y\)
A	\(-1\)	\(2\)
D	\(1\)	\(2\)
B	\(-3\)	\(0\)
C	\(3\)	\(0\)
M	\(- \sqrt{5}\)	\(3 - \sqrt{5}\)
N	\(\sqrt{5}\)	\(3 - \sqrt{5}\)

Certified Golden Ratio: AM / MB = GoldenRatio.