GC-02.04 · Symbolic Model & Source

Companion symbolic Python construction for GC-02.04 · Tran Quang Hung’s Reflected Equilateral Triangles (GC-02.04).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-02.04")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_tran_reflected_equilaterals() -> ConstructionResult:
            """Tran Quang Hung's reflected equilateral triangles & midline secant."""
            m = new_catalog_model("tran-reflected-equilaterals")
            s3 = sp.sqrt(3)
            A = m.set_point(-1, 0, classes=["given"], ID="A")
            B = m.set_point(1, 0, classes=["given"], ID="B")
            C = m.set_point(0, s3, ID="C")
            C1 = m.set_point(0, -s3, ID="C1")
            O1 = m.set_point(0, -s3 / 3, ID="O1")

            m.construct_polygon([A, B, C], ID="ABC")
            m.construct_polygon([A, B, C1], ID="ABC1")
            c_abc1 = m.construct_circle(O1, A, classes=["primary"], ID="cO1")

            N = m.set_point(0, 0, ID="N")
            M = m.set_point(sp.Rational(1, 2), s3 / 2, ID="M")
            ln_mn = m.construct_line(M, N, classes=["auxiliary"], ID="MN")
            P = m.intersect(ln_mn, c_abc1, side_of=(A, B, 1), ID="P")

            ln_ap = m.construct_line(A, P, classes=["golden"], ID="AP")
            ln_bc = m.construct_line(B, C, ID="BC")
            m.intersect(ln_ap, ln_bc, classes=["golden"], ID="D")

            return make_result(m, ("C", "D"), ("D", "B"), ("C", "D", "B"))
        

Exact Symbolic Point Coordinates

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

Point	Exact \(x\)	Exact \(y\)
A	\(-1\)	\(0\)
B	\(1\)	\(0\)
C	\(0\)	\(\sqrt{3}\)
C1	\(0\)	\(- \sqrt{3}\)
O1	\(0\)	\(- \frac{\sqrt{3}}{3}\)
N	\(0\)	\(0\)
M	\(\frac{1}{2}\)	\(\frac{\sqrt{3}}{2}\)
P	\(- \frac{1}{4} + \frac{\sqrt{5}}{4}\)	\(\frac{- \sqrt{3} + \sqrt{15}}{4}\)
D	\(- \frac{1}{2} + \frac{\sqrt{5}}{2}\)	\(\frac{- \sqrt{15} + 3 \sqrt{3}}{2}\)

Certified Golden Ratio: CD / DB = GoldenRatio.