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X₁¶

INCENTER

X₁ is the point of concurrence of the interior angle bisectors of ABC; the point inside ABC whose distances from sidelines BC, CA, AB are equal. This equal distance, r, is the radius of the incircle, given by \(r = 2 \operatorname{area}(ABC)/(a + b + c)\).

X₁ (INCENTER) constructed in exact rational coordinates on the Heronian reference triangle \((a, b, c) = (15, 13, 14)\) over \(\mathbb{Q}\) with vertices A = \((0, 0)\), B = \((14, 0)\), C = \((5, 12)\).¶

"""Exact symbolic GEOMETOR construction model for ``X_1 = INCENTER`` (Euclid IV.4)."""

        from __future__ import annotations

        from pathlib import Path

        import sympy as sp
        import sympy.geometry as spg
        from geometor.model import Model

        from geometor.etc.ops import (
            bevan_point,
            central_line,
            centroid,
            circumcenter,
            construct_baseline_triangle,
            construct_reference_triangle,
            create_model,
            de_longchamps_point,
            feuerbach_point,
            gergonne_point,
            incenter,
            mittenpunkt,
            nagel_point,
            nine_point_center,
            promote_point_to_top,
            spieker_center,
            symmedian_point,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_00001(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct ``X_1 = INCENTER`` and its classical incident central lines on ``ABC``.

            Constructed elements include:

            - Interior angle bisectors ``[ A I_a ]``, ``[ B I_b ]``, ``[ C I_c ]`` concurring at ``X_1``.
            - Right-spread pedal touchpoints ``T_a``, ``T_b``, ``T_c`` and the inscribed ``incircle``.
            - Incident central lines: Nagel ``{ 2, 8 }`` (with ``X_2, X_8, X_10``),
              Incenter-Symmedian ``{ 6, 9 }``, Feuerbach ``{ 5, 11 }``, Soddy ``{ 7, 20 }``,
              and ``OI`` line ``{ 3, 40 }``.
            """
            X_1 = incenter(
                model,
                A,
                B,
                C,
                ID="X_1",
                classes=["selected", "target"],
                draw_bisectors=True,
                draw_incircle=True,
                draw_touchpoints=True,
            )

            X_2 = centroid(model, A, B, C, ID="X_2", classes=["center", "incident"])
            X_8 = nagel_point(model, A, B, C, ID="X_8", classes=["center", "incident"])
            spieker_center(model, A, B, C, ID="X_10", classes=["center", "incident"])
            central_line(model, X_2, X_8, 2, 8, incident=True)

            X_6 = symmedian_point(model, A, B, C, ID="X_6", classes=["center", "incident"])
            X_9 = mittenpunkt(model, A, B, C, ID="X_9", classes=["center", "incident"])
            central_line(model, X_6, X_9, 6, 9, incident=True)

            X_5 = nine_point_center(model, A, B, C, ID="X_5", classes=["center", "incident"])
            X_11 = feuerbach_point(model, A, B, C, ID="X_11", classes=["center", "incident"])
            central_line(model, X_5, X_11, 5, 11, incident=True)

            X_7 = gergonne_point(model, A, B, C, ID="X_7", classes=["center", "incident"])
            X_20 = de_longchamps_point(model, A, B, C, ID="X_20", classes=["center", "incident"])
            central_line(model, X_7, X_20, 7, 20, incident=True)

            X_3 = circumcenter(model, A, B, C, ID="X_3", classes=["center", "incident"])
            X_40 = bevan_point(model, A, B, C, ID="X_40", classes=["center", "incident"])
            central_line(model, X_3, X_40, 3, 40, incident=True)

            return promote_point_to_top(model, X_1)


        def build_model(
            preset: str | tuple[sp.Expr | int, sp.Expr | int, sp.Expr | int] = "heronian",
            show_guides: bool = False,
        ) -> Model:
            """Build an exact symbolic :class:`geometor.model.Model` for ``X_1 = INCENTER``."""
            model = create_model("X_1 = INCENTER")
            if isinstance(preset, str):
                A, B, C = construct_reference_triangle(model, preset, show_guides=show_guides)
            else:
                A, B, C = construct_baseline_triangle(model, *preset, show_guides=show_guides)
            construct_x_00001(model, A, B, C)
            return model


        if __name__ == "__main__":
            here = Path(__file__).resolve().parent
            plate_specs = [
                ("heronian", "x-00001.svg", "Heronian (15, 13, 14) | Δ = 84, r = 4"),
                ("right", "x-00001-right.svg", "Right (3, 4, 5) | Δ = 6, r = 1"),
                ("kepler", "x-00001-kepler.svg", "Kepler (1, √φ, φ) | Δ = √φ / 2"),
                ("kimberling", "x-00001-kimberling.svg", "Kimberling (6, 9, 13) | r = 2√35 / 7"),
            ]
            for preset_key, svg_name, sub_text in plate_specs:
                mdl = build_model(preset_key)
                if preset_key == "heronian":
                    mdl.save(str(here / "x-00001.json"))
                write_center_svg(mdl, here / svg_name, subtitle=sub_text)
        

X1¶

X₁¶