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X₁₁₄₅¶

3rd EHRMANN POINT

Let J_a,J_b,J_c be the respective excenters of ABC, and let A_B be the projection of A on J_aJ_b, let A_C be the projection of A on J_aJ_c, and define B_C, B_A, C_A, C_B cyclically. The Euler lines of the three triangles J_aA_BA_C, J_bB_CB_A, J_cC_AC_B concur in X₁₁₄₅. Also, X₁₁₄₅ is X₉₇₄ of the excentral triangle. (Analogously, X₄₄₂ is X₉₇₃ of the excentral triangle; see the note at X₄₄₂.) Jean-Pierre Ehrmann (#4200, 10/24/01)

X₁₁₄₅ (3rd EHRMANN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1145 = 3rd EHRMANN 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,
            circumcenter_of,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            feuerbach_point_of,
            midpoint,
            mittenpunkt_of,
            nagel_point_of,
            promote_point_to_top,
            spieker_center_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01145(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1145 = 3rd EHRMANN POINT on reference triangle ABC."""
            X_8 = nagel_point_of(model, A, B, C, ID="X_8", classes=["center"])
            X_100 = eval_trilinear_point(model, A, B, C, '1/(b - c)', ID="X_100", classes=["center"])
            X_1145 = midpoint(model, X_8, X_100, ID="X_1145", classes=["selected", "target"], guide=False)
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["center", "incident"])
            central_line(model, X_3, X_8, 3, 8, incident=True)
            X_10 = spieker_center_of(model, A, B, C, ID="X_10", classes=["center", "incident"])
            X_11 = feuerbach_point_of(model, A, B, C, ID="X_11", classes=["center", "incident"])
            central_line(model, X_10, X_11, 10, 11, incident=True)
            X_9 = mittenpunkt_of(model, A, B, C, ID="X_9", classes=["center", "incident"])
            X_80 = eval_trilinear_point(model, A, B, C, '1/(1 - 2*cos(A))', ID="X_80", classes=["center", "incident"])
            central_line(model, X_9, X_80, 9, 80, incident=True)
            return promote_point_to_top(model, X_1145)


        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_1145 on the given reference triangle."""
            model = create_model('X_1145 = 3rd EHRMANN POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01145(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-01145.json"))
            write_center_svg(
                model_acute,
                here / "x-01145.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1145¶

X₁₁₄₅¶