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X₁₁₂₃¶

PAASCHE POINT

Let D and E be the congruent circles each tangent to the other and to line BC, with D also tangent to line AB and E also tangent to line CA, meeting in a point A' lying outside triangle ABC. Define B' and C' cyclically. Then A'B'C' is perspective to ABC, and the perspector is X₁₁₂₃. See

X₁₁₂₃ (PAASCHE POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_1123 = PAASCHE 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,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_01123(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_1123 = PAASCHE POINT on reference triangle ABC."""
            X_1123 = eval_trilinear_point(model, A, B, C, '1/(1 +sin(A))', ID="X_1123", classes=["selected", "target"])
            X_37 = eval_trilinear_point(model, A, B, C, 'b + c', ID="X_37", classes=["center", "incident"])
            X_158 = eval_trilinear_point(model, A, B, C, '(sec(A))**(2)', ID="X_158", classes=["center", "incident"])
            central_line(model, X_37, X_158, 37, 158, incident=True)
            X_57 = eval_trilinear_point(model, A, B, C, '1/(b + c - a)', ID="X_57", classes=["center", "incident"])
            X_482 = eval_trilinear_point(model, A, B, C, '1 + 2*sec(A/2)*cos(B/2)*cos(C/2)', ID="X_482", classes=["center", "incident"])
            central_line(model, X_57, X_482, 57, 482, incident=True)
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_586 = eval_trilinear_point(model, A, B, C, '((2*area*(1/b + 1/c - 1/a) + a - b - c)/a)', ID="X_586", classes=["center", "incident"])
            central_line(model, X_2, X_586, 2, 586, incident=True)
            return promote_point_to_top(model, X_1123)


        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_1123 on the given reference triangle."""
            model = create_model('X_1123 = PAASCHE POINT')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_01123(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-01123.json"))
            write_center_svg(
                model_acute,
                here / "x-01123.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X1123¶

X₁₁₂₃¶