← etc / Encyclopedia of Triangle Centers

X₃₅₇₁¶

INTERSECTION OF LINES P(6)P(8) AND U(6)U(8)

The points given by trilinears P(6) =\(b : c : a\) and U(6) =\(c : a : b\) are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) =\(1/b : 1/c : 1/a\) and U(8) =\(1/c : 1/a : 1/b\). The bicentric pair of lines P(6)P(8) and U(6)U(8) concur in X₃₅₇₁. Contributed by Peter Moses, July 7, 2009.

X₃₅₇₁ (INTERSECTION OF LINES P(6)P(8) AND U(6)U(8)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3571 = INTERSECTION OF LINES P(6)P(8) AND U(6)U(8)."""

        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,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            incenter_of,
            mittenpunkt_of,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_03571(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3571 = INTERSECTION OF LINES P(6)P(8) AND U(6)U(8) on reference triangle ABC."""
            X_3571 = eval_trilinear_point(model, A, B, C, 'a**(4)*b*c + a**(2)*(b**(4) - b**(3)*c - 2*b**(2)*c**(2) - b*c**(3) + c**(4)) + b**(3)*c**(3)', ID="X_3571", classes=["selected", "target"])
            X_9 = mittenpunkt_of(model, A, B, C, ID="X_9", classes=["center", "incident"])
            X_43 = eval_trilinear_point(model, A, B, C, 'a*b + a*c - b*c', ID="X_43", classes=["center", "incident"])
            central_line(model, X_9, X_43, 9, 43, incident=True)
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_512 = eval_trilinear_point(model, A, B, C, 'a*(b**(2) - c**(2))', ID="X_512", classes=["center", "incident"])
            central_line(model, X_1, X_512, 1, 512, incident=True)
            X_1621 = eval_trilinear_point(model, A, B, C, '(a**(2) - b*c - c*a - a*b)', ID="X_1621", classes=["center", "incident"])
            X_1964 = eval_trilinear_point(model, A, B, C, 'a**(2)*(b**(2) + c**(2))', ID="X_1964", classes=["center", "incident"])
            central_line(model, X_1621, X_1964, 1621, 1964, incident=True)
            return promote_point_to_top(model, X_3571)


        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_3571 on the given reference triangle."""
            model = create_model('X_3571 = INTERSECTION OF LINES P(6)P(8) AND U(6)U(8)')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03571(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-03571.json"))
            write_center_svg(
                model_acute,
                here / "x-03571.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3571¶

X₃₅₇₁¶