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X₃₆₄₉¶

KS(INTOUCH TRIANGLE)

Two proofs of Dao’s theorem about cyclic hexagons were published in 2014: Nikolaos Dergiades’s Dao’s Theorem on Six Circumcenters Associated with a Cyclic Hexagon and Telv Cohl’s A Purely Synthetic Proof of Dao’s Theorem on Six Circumcenters Associated with a Cyclic Hexagon. In the degenerate case that the hexagon is the intouch triangle A'B'C', the triangle of the circumcenters of AB'C', A'BC', A'B'C is perspective to A'B'C' with perspector X₃₆₄₉. (Dao Thanh Oai, Francisco Javier Garcia Capitan, ADGEOM #1718, September 16, 2014)

X₃₆₄₉ (KS(INTOUCH TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

"""GEOMETOR construction model for X_3649 = KS(INTOUCH TRIANGLE)."""

        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,
            gergonne_point_of,
            incenter_of,
            promote_point_to_top,
            spieker_center_of,
        )
        from geometor.etc.render import write_center_svg


        def construct_x_03649(
            model: Model, A: spg.Point, B: spg.Point, C: spg.Point
        ) -> spg.Point:
            """Construct X_3649 = KS(INTOUCH TRIANGLE) on reference triangle ABC."""
            X_3649 = eval_trilinear_point(model, A, B, C, '(b*c*(b + c)*(2*a + b + c)*(a - b + c)*(a + b - c))', ID="X_3649", classes=["selected", "target"])
            X_10 = spieker_center_of(model, A, B, C, ID="X_10", classes=["center", "incident"])
            X_12 = eval_trilinear_point(model, A, B, C, '1 + cos(B - C)', ID="X_12", classes=["center", "incident"])
            central_line(model, X_10, X_12, 10, 12, 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_1 = incenter_of(model, A, B, C, ID="X_1", classes=["center", "incident"])
            X_30 = eval_trilinear_point(model, A, B, C, 'cos(A) - 2*cos(B)*cos(C)', ID="X_30", classes=["center", "incident"])
            central_line(model, X_1, X_30, 1, 30, incident=True)
            return promote_point_to_top(model, X_3649)


        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_3649 on the given reference triangle."""
            model = create_model('X_3649 = KS(INTOUCH TRIANGLE)')
            A, B, C = construct_baseline_triangle(model, *sides, show_guides=show_guides)
            construct_x_03649(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-03649.json"))
            write_center_svg(
                model_acute,
                here / "x-03649.svg",
                subtitle="(a, b, c) = (5, 6, 7) | AB = 1",
            )
        

X3649¶

X₃₆₄₉¶