← etc / Encyclopedia of Triangle Centers

X₃₃₆₃¶

X₆ OF PEDAL TRIANGLE OF X₂

X₃₃₆₃ is the symmedian point of the pedal triangle of the centroid. A second construction uses the midpoint X₅₉₇ of the symmedian point and centroid and the midpoint X₁₁₅ of the two Fermat points, X₁₃ and X₁₄; specifically, X₃₃₆₃ is the point in which the line X₁₁₅ X₅₉₇ meets the Euler line. Contributed by Po-chieh Chen and Shao-cheng Liu, May 29, 2008.

X₃₃₆₃ (X₆ OF PEDAL TRIANGLE OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3363 = X_6 OF PEDAL TRIANGLE OF X_2."""

        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,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_03363(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3363 = X_6 OF PEDAL TRIANGLE OF X_2 on reference triangle ABC."""
            X_3363 = eval_trilinear_point(model, A, B, C, '(b*c*(-4*a**(4) + 5*b**(4) + 5*c**(4) - 14*b**(2)*c**(2) - 5*c**(2)*a**(2) - 5*a**(2)*b**(2)))', ID="X_3363", 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_115 = eval_trilinear_point(model, A, B, C, 'b*c*(b**(2) - c**(2))**(2)', ID="X_115", classes=["center", "incident"])
            X_597 = eval_trilinear_point(model, A, B, C, '(4*a**(2) + b**(2) + c**(2))/a', ID="X_597", classes=["center", "incident"])
            central_line(model, X_115, X_597, 115, 597, incident=True)
            return promote_point_to_top(model, X_3363)


        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_3363 on the given reference triangle."""
            model = create_model('X_3363 = X_6 OF PEDAL TRIANGLE OF X_2')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03363(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-03363.json"))
            write_center_svg(
                model_acute,
                here / "x-03363.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3363¶

X₃₃₆₃¶