G.85 square of a sum — Code

II.4

If a segment is cut at random, the square on the whole equals the squares on the parts together with twice the rectangle contained by the parts.

"""Programmatic construction for Book II, Proposition 4 (II.4).

        If a straight line be cut at random, the square on the whole is equal to the
        squares on the segments and twice the rectangle contained by the segments.
        """

        from __future__ import annotations

        from geometor.model import Model


        def construct_g1() -> Model:
            """Construct the figure for Book II, Proposition 4 using geometor.model."""
            model = Model("II.4")

            # Straight line AB cut at random at C
            A = model.set_point(28.5, -29.5, classes=["given"], ID="A")
            C = model.set_point(124.5, -29.5, classes=["given"], ID="C")
            B = model.set_point(170.5, -29.5, classes=["given"], ID="B")

            model.construct_line(A, B)
            model.set_segment(A, B)
            model.set_segment(A, C)
            model.set_segment(C, B)

            # Square ADEB described on AB [I. 46], diagonal BD joined
            D = model.set_point(28.5, -171.5, ID="D")
            E = model.set_point(170.5, -171.5, ID="E")

            model.set_segment(A, D)
            model.set_segment(D, E)
            model.set_segment(B, E)
            model.set_polygon([A, D, E, B])

            model.construct_line(B, D)
            model.set_segment(B, D)

            # Parallel CF to AD/EB through C, meeting BD at G and DE at F [I. 31]
            G = model.set_point(124.5, -75.5, ID="G")
            F = model.set_point(124.5, -171.5, ID="F")
            model.construct_line(C, F)
            model.set_segment(C, F)
            model.set_segment(C, G)
            model.set_segment(G, F)
            model.set_segment(D, F)
            model.set_segment(F, E)
            model.set_segment(B, G)
            model.set_segment(G, D)

            # Parallel HK to AB/DE through G, meeting AD at H and BE at K [I. 31]
            H = model.set_point(28.5, -75.5, ID="H")
            K = model.set_point(170.5, -75.5, ID="K")
            model.construct_line(H, K)
            model.set_segment(H, K)
            model.set_segment(H, G)
            model.set_segment(G, K)
            model.set_segment(A, H)
            model.set_segment(H, D)
            model.set_segment(B, K)
            model.set_segment(K, E)

            # Squares about diameter: CGKB (CK) on CB and HDFG (HF) on AC
            model.set_polygon([C, G, K, B])
            model.set_polygon([H, D, F, G])

            # Complements AG (ACGH) and GE (GKEF), plus composite rectangles AF (ACFD) and FK (HKEF)
            model.set_polygon([A, C, G, H])
            model.set_polygon([G, K, E, F])
            model.set_polygon([A, C, F, D])
            model.set_polygon([H, K, E, D])

            # Gnomon around CK: A C G K E D
            model.set_polygon([A, C, G, K, E, D])

            # Referenced triangles ABD, CGB, GDF
            model.set_polygon([A, B, D])
            model.set_polygon([C, G, B])
            model.set_polygon([H, G, D])

            # Referenced angles CGB, ADB, ABD, GBC, KBC, GCB, CGK, GKB
            model.set_wedge(G, C, C, B)  # < C G B )
            model.set_wedge(D, A, A, B)  # < A D B )
            model.set_wedge(B, A, A, D)  # < A B D ) / < G B C )
            model.set_wedge(B, K, K, C)  # < K B C )
            model.set_wedge(C, G, G, B)  # < G C B )
            model.set_wedge(G, C, C, K)  # < C G K )
            model.set_wedge(K, G, G, B)  # < G K B )

            return model


        if __name__ == "__main__":
            construct_g1()