GC-06.04 · Symbolic Model & Source

Companion symbolic Python construction for GC-06.04 · Tran Quang Hung & Elliot McGucken’s Three Tangent Circles (GC-06.04).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-06.04")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_tran_mcgucken_three_tangent_circles() -> ConstructionResult:
            """Tran Quang Hung & Elliot McGucken's three mutually tangent equal circles."""
            m = new_catalog_model("tran-mcgucken-three-tangent-circles")
            s3 = sp.sqrt(3)

            O = m.set_point(0, 0, classes=["given"], ID="O")
            A = m.set_point(0, 2 * s3 / 3, classes=["given"], ID="A")
            B = m.set_point(-1, -s3 / 3, ID="B")
            C = m.set_point(1, -s3 / 3, ID="C")

            D = m.set_point(0, -s3 / 3, classes=["golden"], ID="D")
            E = m.set_point(sp.Rational(1, 2), s3 / 6, ID="E")
            F = m.set_point(-sp.Rational(1, 2), s3 / 6, classes=["golden"], ID="F")
            _ = E

            m.construct_polygon([A, B, C], classes=["auxiliary"], ID="ABC")
            m.construct_circle(A, F, classes=["compass"], ID="cA")
            m.construct_circle(B, D, classes=["compass"], ID="cB")
            m.construct_circle(C, D, classes=["compass"], ID="cC")
            c_o = m.construct_circle(O, A, classes=["primary"], ID="cO")

            ln_df = m.construct_line(D, F, classes=["golden"], ID="DF")
            m.intersect(c_o, ln_df, beyond=(D, F), classes=["golden"], ID="G")
            J = m.intersect(c_o, ln_df, beyond=(F, D), classes=["golden"], ID="J")
            _ = J

            return make_result(m, ("D", "F"), ("F", "G"), ("D", "F", "G"))
        

Exact Symbolic Point Coordinates

All points in tran-mcgucken-three-tangent-circles are certified free of floating-point approximations over \(\mathbb{Q}(\sqrt{3}, \phi)\):

Point	Exact \(x\)	Exact \(y\)
O	\(0\)	\(0\)
A	\(0\)	\(\frac{2 \sqrt{3}}{3}\)
B	\(-1\)	\(- \frac{\sqrt{3}}{3}\)
C	\(1\)	\(- \frac{\sqrt{3}}{3}\)
D	\(0\)	\(- \frac{\sqrt{3}}{3}\)
E	\(\frac{1}{2}\)	\(\frac{\sqrt{3}}{6}\)
F	\(- \frac{1}{2}\)	\(\frac{\sqrt{3}}{6}\)
G	\(- \frac{\sqrt{5}}{4} - \frac{1}{4}\)	\(\frac{- \sqrt{3} + 3 \sqrt{15}}{12}\)
J	\(- \frac{1}{4} + \frac{\sqrt{5}}{4}\)	\(\frac{- 3 \sqrt{15} - \sqrt{3}}{12}\)

Certified Golden Ratio: DF / FG = GoldenRatio.