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X₄₂₈¶

EULER X₃₈-5th-SUBSTITUTION POINT

Let H = X₄ and let H_aH_bH_c be the pedal triangle of H in the plane of a triangle ABC. Let

B_c = reflection of H_b in HH_c, and define C_a and A_b cyclically;

C_b = reflection of H_c in HH_b, and define A_c and B_a cyclically;

H_a = orthogonal projecton of A on B_cC_b, and define H_b and H_c cyclically.

Then X₄₂₈ = centroid of triangle H_aH_bH_c. See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27169

X₄₂₈ (EULER X₃₈-5th-SUBSTITUTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_428 = EULER X_38-5th-SUBSTITUTION POINT."""

        from __future__ import annotations

        from pathlib import Path

        import sympy.geometry as spg
        from geometor.model import Model

        from geometor.etc.ops import (
            central_line,
            centroid_of,
            circumcenter_of,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            point_from_barycentrics,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00428(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_428 = EULER X_38-5th-SUBSTITUTION POINT on reference triangle ABC."""
            X_428 = eval_trilinear_point(model, A, B, C, '3*sec(A) - csc(A)*tan(omega) ', ID="X_428", classes=["selected", "target"])
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center"])
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["center"])
            central_line(model, X_2, X_3, 2, 3)
            X_132 = point_from_barycentrics(model, A, B, C, 0.11002416, 0.049913231, 0.84006261, ID="X_132", classes=["center", "incident"])
            X_137 = point_from_barycentrics(model, A, B, C, 0.1354431, 0.64976082, 0.21479608, ID="X_137", classes=["center", "incident"])
            central_line(model, X_132, X_137, 132, 137, incident=True)
            X_51 = eval_trilinear_point(model, A, B, C, 'a**(2)*cos(B - C)', ID="X_51", classes=["center", "incident"])
            X_1503 = eval_trilinear_point(model, A, B, C, 'sin(A)*tan(omega)  - 2*cos(B)*cos(C)', ID="X_1503", classes=["center", "incident"])
            central_line(model, X_51, X_1503, 51, 1503, incident=True)
            return promote_point_to_top(model, X_428)


        def build_model(
            sides: tuple[float, float, float] = (5.0, 6.0, 7.0),
            show_guides: bool = False,
        ) -> Model:
            """Build a `geometor.model.Model` for X_428 on the given reference triangle."""
            model = create_model('X_428 = EULER X_38-5th-SUBSTITUTION POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00428(model, A, B, C)
            return model


        if __name__ == "__main__":
            here = Path(__file__).resolve().parent
            model_acute = build_model((5.0, 6.0, 7.0))
            model_acute.save(str(here / "x-00428.json"))
            write_center_svg(
                model_acute,
                here / "x-00428.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X428¶

X₄₂₈¶