GC-07.05 · Symbolic Model & Source

Companion symbolic Python construction for GC-07.05 · Tran Quang Hung’s Mixtilinear Circle in an Equilateral Triangle (GC-07.05).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-07.05")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_tran_mixtilinear_equilateral() -> ConstructionResult:
            """Tran Quang Hung's Golden Ratio in a mixtilinear circle of an equilateral triangle."""
            m = new_catalog_model("tran-mixtilinear-equilateral")
            s3 = sp.sqrt(3)

            # Side 18 equilateral triangle ABC => all key points have exact integer * radical coordinates
            A = m.set_point(-9, 0, classes=["given"], ID="A")
            B = m.set_point(9, 0, classes=["given"], ID="B")
            C = m.set_point(0, 9 * s3, ID="C")
            K = m.set_point(0, 3 * s3, ID="K")

            m.construct_polygon([A, B, C], ID="ABC")
            m.construct_circle(K, C, classes=["primary"], ID="cK")

            # A-mixtilinear incircle centered at F=(3, 4*sqrt(3)), tangent to AB at D=(3,0) and (ABC) at T=(9, 6*sqrt(3))
            D = m.set_point(3, 0, ID="D")
            F = m.set_point(3, 4 * s3, ID="F")
            c_f = m.construct_circle(F, D, classes=["golden"], ID="cF")

            # Horizontal line through circumcenter K=(0, 3*sqrt(3)) meets BC at O=(6, 3*sqrt(3))
            O = m.set_point(6, 3 * s3, classes=["golden"], ID="O")
            ln_ko = m.construct_line(K, O, classes=["golden"], ID="KO")

            I = m.intersect(c_f, ln_ko, beyond=(O, K), classes=["golden"], ID="I")
            m.intersect(c_f, ln_ko, beyond=(K, O), classes=["golden"], ID="J")
            _ = I

            return make_result(m, ("K", "O"), ("O", "J"), ("K", "O", "J"))
        

Exact Symbolic Point Coordinates

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

Point	Exact \(x\)	Exact \(y\)
A	\(-9\)	\(0\)
B	\(9\)	\(0\)
C	\(0\)	\(9 \sqrt{3}\)
K	\(0\)	\(3 \sqrt{3}\)
D	\(3\)	\(0\)
F	\(3\)	\(4 \sqrt{3}\)
O	\(6\)	\(3 \sqrt{3}\)
I	\(3 - 3 \sqrt{5}\)	\(3 \sqrt{3}\)
J	\(3 + 3 \sqrt{5}\)	\(3 \sqrt{3}\)

Certified Golden Ratio: KO / OJ = GoldenRatio.