GC-01.02 · Symbolic Model & Source

Companion symbolic Python construction for GC-01.02 · Right-Triangle Hypotenuse Transfer (1 : 2 : √5) (GC-01.02).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-01.02")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_pythagorean_transfer() -> ConstructionResult:
            """Heron-Ptolemy right-triangle hypotenuse transfer on a 1:2 right triangle."""
            m = new_catalog_model("pythagorean-transfer")
            A = m.set_point(0, 0, classes=["given"], ID="A")
            B = m.set_point(2, 0, classes=["given"], ID="B")
            M = m.set_point(1, 0, ID="M")
            C = m.set_point(2, 1, ID="C")

            seg_ab = m.construct_segment(A, B, ID="AB")
            m.construct_segment(B, C, ID="BC")
            ln_ac = m.construct_line(A, C, classes=["auxiliary"], ID="AC")
            m.construct_polygon([A, B, C], ID="ABC")

            c_bm = m.construct_circle(B, M, classes=["auxiliary"], ID="cBM")
            _ = c_bm
            c_cb = m.construct_circle(C, B, classes=["compass"], ID="cCB")
            D = m.intersect(c_cb, ln_ac, between=(A, C), ID="D")
            E = m.intersect(c_cb, ln_ac, beyond=(A, C), ID="E")

            c_ad = m.construct_circle(A, D, classes=["golden"], ID="cAD")
            m.intersect(c_ad, seg_ab, between=(A, B), classes=["golden"], ID="G")
            _ = E

            return make_result(m, ("A", "G"), ("G", "B"), ("A", "G", "B"))
        

Exact Symbolic Point Coordinates

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

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

Certified Golden Ratio: AG / GB = GoldenRatio.