← etc / Encyclopedia of Triangle Centers

X₂₀₉₂¶

DANNEELS-APOLLONIUS PERSPECTOR

Let A' be the point common to Apollonius circle and the A-excircle, and define B' and C' cyclically. Then triangle A'B'C' is perspective to the cevian triangle of X₆, and the perspector is X₂₀₉₂. (Eric Danneels, Hyacinthos, #8070, 10/01/03). (Triangle A'B'C' is also perspective to the cevian triangle of X₁; see X₁₆₈₂.)

X₂₀₉₂ (DANNEELS-APOLLONIUS PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_2092 = DANNEELS-APOLLONIUS PERSPECTOR."""

        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,
            complement,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            mittenpunkt_of,
            promote_point_to_top,
            spieker_center_of,
            symmedian_point_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_02092(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_2092 = DANNEELS-APOLLONIUS PERSPECTOR on reference triangle ABC."""
            X_314 = eval_trilinear_point(model, A, B, C, '(b**(2)*c**(2)*(b + c - a)/(b + c))', ID="X_314", classes=["center"])
            X_2092 = complement(model, A, B, C, X_314, ID="X_2092", classes=["selected", "target"])
            X_3 = circumcenter_of(model, A, B, C, ID="X_3", classes=["center", "incident"])
            X_6 = symmedian_point_of(model, A, B, C, ID="X_6", classes=["center", "incident"])
            central_line(model, X_3, X_6, 3, 6, incident=True)
            X_10 = spieker_center_of(model, A, B, C, ID="X_10", classes=["center", "incident"])
            X_37 = eval_trilinear_point(model, A, B, C, 'b + c', ID="X_37", classes=["center", "incident"])
            central_line(model, X_10, X_37, 10, 37, incident=True)
            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)
            return promote_point_to_top(model, X_2092)


        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_2092 on the given reference triangle."""
            model = create_model('X_2092 = DANNEELS-APOLLONIUS PERSPECTOR')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_02092(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-02092.json"))
            write_center_svg(
                model_acute,
                here / "x-02092.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X2092¶

X₂₀₉₂¶