← etc / Encyclopedia of Triangle Centers

X₁₉₉₁¶

2nd VAN LAMOEN PERPENDICULAR BISECTORS POINT

Erect squares inwardly on the sides of triangle ABC. Two edges emanate from A; let P and Q be their endpoints. Let a’ be the perpendicular bisector of PQ, and define b’ and c’ cyclically. Then a’, b’, c’ concur in X₁₉₉₁. See X₅₉₁ for the 1st Van Lamoen perpendicular bisectors point, constructed from outwardly drawn squares.

X₁₉₉₁ (2nd VAN LAMOEN PERPENDICULAR BISECTORS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1991 = 2nd VAN LAMOEN PERPENDICULAR BISECTORS 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,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            point_from_barycentrics,
            promote_point_to_top,
            reflection_in_point,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01991(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1991 = 2nd VAN LAMOEN PERPENDICULAR BISECTORS POINT on reference triangle ABC."""
            X_591 = eval_trilinear_point(model, A, B, C, 'b*c*(b**(2) + c**(2) - 2*a**(2) + 4*area )', ID="X_591", classes=["center"])
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center"])
            X_1991 = reflection_in_point(model, X_591, X_2, ID="X_1991", classes=["selected", "target"], guide=False)
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            central_line(model, X_2, X_6, 2, 6, incident=True)
            X_371 = eval_trilinear_point(model, A, B, C, 'cos(A -  pi  /4)', ID="X_371", classes=["center", "incident"])
            X_754 = eval_trilinear_point(model, A, B, C, '(a **(-1)*(b**(4) + c**(4)) - a**(3)*(b**(0) + c**(0)))', ID="X_754", classes=["center", "incident"])
            central_line(model, X_371, X_754, 371, 754, incident=True)
            X_638 = point_from_barycentrics(model, A, B, C, 1.0201326, 0.50285924, -0.52299185, ID="X_638", classes=["center", "incident"])
            X_1151 = eval_trilinear_point(model, A, B, C, 'sin(A) + 2*cos(A)', ID="X_1151", classes=["center", "incident"])
            central_line(model, X_638, X_1151, 638, 1151, incident=True)
            return promote_point_to_top(model, X_1991)


        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_1991 on the given reference triangle."""
            model = create_model('X_1991 = 2nd VAN LAMOEN PERPENDICULAR BISECTORS POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01991(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-01991.json"))
            write_center_svg(
                model_acute,
                here / "x-01991.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1991¶

X₁₉₉₁¶