GC-07.01 · Symbolic Model & Source

Companion symbolic Python construction for GC-07.01 · Tran Quang Hung’s 2:1 Rhombus & Incircle Diagonal Division (GC-07.01).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-07.01")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_tran_rhombus_incircle() -> ConstructionResult:
            """Tran Quang Hung's rhombus with diagonal ratio 2:1 and its incircle (Buratino 8)."""
            m = new_catalog_model("tran-rhombus-incircle")
            s5 = sp.sqrt(5)

            O = m.set_point(0, 0, classes=["given"], ID="O")
            A = m.set_point(0, s5, ID="A")
            C = m.set_point(0, -s5, ID="C")
            B = m.set_point(2 * s5, 0, classes=["given"], ID="B")
            D = m.set_point(-2 * s5, 0, ID="D")

            m.construct_polygon([A, B, C, D], ID="ABCD")
            m.construct_segment(A, C, classes=["auxiliary"], ID="AC")
            ln_bd = m.construct_line(D, B, classes=["golden"], ID="DB")

            # Inradius r = (2*sqrt(5) * sqrt(5)) / 5 = 2; tangency on AB at H = (2/sqrt(5), 4/sqrt(5))
            F = m.set_point(2, 0, classes=["golden"], ID="F")
            c_o = m.construct_circle(O, F, classes=["golden"], ID="cO")
            m.intersect(c_o, ln_bd, other_than=F, classes=["golden"], ID="E")

            return make_result(m, ("E", "F"), ("F", "B"), ("E", "F", "B"))
        

Exact Symbolic Point Coordinates

All points in tran-rhombus-incircle are certified free of floating-point approximations over \(\mathbb{Q}(\phi)\):

Point	Exact \(x\)	Exact \(y\)
O	\(0\)	\(0\)
A	\(0\)	\(\sqrt{5}\)
C	\(0\)	\(- \sqrt{5}\)
B	\(2 \sqrt{5}\)	\(0\)
D	\(- 2 \sqrt{5}\)	\(0\)
F	\(2\)	\(0\)
E	\(-2\)	\(0\)

Certified Golden Ratio: EF / FB = GoldenRatio.