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X₃₅₅¶

FUHRMANN CENTER

X₃₅₅ = the center of the Fuhrmann circle, defined as the circumcircle of the Fuhrmann triangle A″B″C″, where A″ is obtained as follows: let A' be the midpoint of the circumcircle-arc having endpoints B and C and not containing A; then A″ is the reflection of A' in line BC. Vertices B″ and C″ are obtained cyclically. (Other constructions of A″, and hence the Fuhrmann triangle, follow: (1) Let I_A be the reflection of X₁ in BC; then A″ is the circumcenter of I_ABC. (2) Let J_A be the reflection of the A-excenter in BC; then A″ is the circumcenter of J_ABC.

X₃₅₅ (FUHRMANN CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_355 = FUHRMANN CENTER."""

        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,
            circumcenter_of,
            construct_baseline_triangle,
            create_model,
            incenter_of,
            midpoint,
            nagel_point_of,
            nine_point_center_of,
            orthocenter_of,
            promote_point_to_top,
            spieker_center_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00355(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_355 = FUHRMANN CENTER on reference triangle ABC."""
            X_4 = orthocenter_of(model, A, B, C, ID="X_4", classes=["center"])
            X_8 = nagel_point_of(model, A, B, C, ID="X_8", classes=["center"])
            X_355 = midpoint(model, X_4, X_8, ID="X_355", classes=["selected", "target"], guide=False)
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_5 = nine_point_center_of(model, A, B, C, ID="X_5", classes=["center", "incident"])
            central_line(model, X_1, X_5, 1, 5, incident=True)
            central_line(model, X_4, X_8, 4, 8, incident=True)
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["center", "incident"])
            X_10 = spieker_center_of(model, A, B, C, ID="X_10", classes=["center", "incident"])
            central_line(model, X_3, X_10, 3, 10, incident=True)
            return promote_point_to_top(model, X_355)


        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_355 on the given reference triangle."""
            model = create_model('X_355 = FUHRMANN CENTER')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_00355(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-00355.json"))
            write_center_svg(
                model_acute,
                here / "x-00355.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X355¶

X₃₅₅¶