GC-04.04 · Symbolic Model & Source

Companion symbolic Python construction for GC-04.04 · Bùi Quang Tuân’s Van Obel Division in a Right Isosceles Triangle (GC-04.04).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-04.04")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_bui_van_obel_isosceles() -> ConstructionResult:
            """Bùi Quang Tuân's Van Obel cevian division in a right isosceles triangle."""
            m = new_catalog_model("bui-van-obel-isosceles")
            B = m.set_point(-2, 0, classes=["given"], ID="B")
            C = m.set_point(2, 0, classes=["given"], ID="C")
            A = m.set_point(0, 2, ID="A")
            M = m.set_point(0, 0, ID="M")
            O = m.set_point(0, 1, ID="O")

            m.construct_polygon([A, B, C], ID="ABC")
            ln_am = m.construct_line(A, M, classes=["golden"], ID="AM")
            c_om = m.construct_circle(O, M, classes=["primary"], ID="cOM")

            ln_oc = m.construct_line(O, C, classes=["auxiliary"], ID="OC")
            ln_ob = m.construct_line(O, B, classes=["auxiliary"], ID="OB")
            D = m.intersect(c_om, ln_oc, between=(O, C), ID="D")
            F = m.intersect(c_om, ln_ob, between=(O, B), ID="F")

            ln_bd = m.construct_line(B, D, classes=["golden"], ID="BD")
            ln_cf = m.construct_line(C, F, classes=["golden"], ID="CF")
            _ = ln_cf
            m.intersect(ln_bd, ln_am, classes=["golden"], ID="E")

            return make_result(m, ("O", "E"), ("E", "M"), ("O", "E", "M"))
        

Exact Symbolic Point Coordinates

All points in bui-van-obel-isosceles are certified free of floating-point approximations over \(\mathbb{Q}(\phi)\):

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

Certified Golden Ratio: OE / EM = GoldenRatio.