← etc / Encyclopedia of Triangle Centers

X₃₆₁₆¶

H(X₁; M, 1/2)

Let O_A be the circle tangent to side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define O_B and O_C cyclically. Let L_A be the external tangent to circles O_B and O_C that is the nearer of the two to O_A. Define L_B and L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. Then A'B'C' is homothetic to ABC, and the center of homothety is X₃₆₁₆. See the reference at X₁₀₀₁.

X₃₆₁₆ (H(X₁; M, 1/2)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3616 = H(X_1; M, 1/2)."""

        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,
            construct_baseline_triangle,
            create_model,
            eval_trilinear_point,
            gergonne_point_of,
            incenter_of,
            isogonal_conjugate,
            point_from_barycentrics,
            promote_point_to_top,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_03616(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3616 = H(X_1; M, 1/2) on reference triangle ABC."""
            X_2334 = point_from_barycentrics(model, A, B, C, 0.2463661, 0.33111604, 0.42251786, ID="X_2334", classes=["center"])
            X_3616 = isogonal_conjugate(model, A, B, C, X_2334, ID="X_3616", classes=["selected", "target"])
            X_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_2 = centroid_of(model, A, B, C, ID="X_2", classes=["center", "incident"])
            central_line(model, X_1, X_2, 1, 2, incident=True)
            X_7 = gergonne_point_of(model, A, B, C, ID="X_7", classes=["center", "incident"])
            X_21 = eval_trilinear_point(model, A, B, C, '1/(cos(B) + cos(C))', ID="X_21", classes=["center", "incident"])
            central_line(model, X_7, X_21, 7, 21, incident=True)
            X_29 = eval_trilinear_point(model, A, B, C, '(sec(A))/(cos(B) + cos(C))', ID="X_29", classes=["center", "incident"])
            X_278 = eval_trilinear_point(model, A, B, C, 'sec(A)*tan(A/2)', ID="X_278", classes=["center", "incident"])
            central_line(model, X_29, X_278, 29, 278, incident=True)
            return promote_point_to_top(model, X_3616)


        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_3616 on the given reference triangle."""
            model = create_model('X_3616 = H(X_1; M, 1/2)')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03616(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-03616.json"))
            write_center_svg(
                model_acute,
                here / "x-03616.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3616¶

X₃₆₁₆¶