"""Programmatic construction for Book I, Proposition 2 (I.2).
To place at a given point (as an extremity) a straight line equal to a given straight line.
"""
from __future__ import annotations
import sympy as sp
from geometor.model import Model
def construct_g1() -> Model:
"""Construct the figure for Book I, Proposition 2 using geometor.model."""
model = Model("I.2")
# 1. Let A be the given point, and BC the given straight line.
A = model.set_point(0, 0, classes=["given"], ID="A")
B = model.set_point(1, 0, classes=["given"], ID="B")
C = model.set_point(sp.Rational(3, 2), sp.sqrt(3), classes=["given"], label="C")
model.set_segment(B, C)
# 2. From A to B let the straight line AB be joined [Post. 1]
model.set_segment(A, B)
# 3. On AB let the equilateral triangle DAB be constructed [I. 1]
D = model.set_point(sp.Rational(1, 2), sp.sqrt(3) / 2, label="D")
model.set_segment(D, A)
model.set_segment(D, B)
model.set_polygon([D, A, B])
# 4. Let the straight lines AE, BF be produced in a straight line with DA, DB [Post. 2]
model.construct_line(D, A)
model.construct_line(D, B)
# 5. With centre B and distance BC let the circle CGH be described [Post. 3]
model.construct_circle(B, C)
# Distance BC = sqrt((3/2 - 1)^2 + (sqrt(3))^2) = sqrt(1/4 + 3) = sqrt(13)/2
r_BC = sp.sqrt(13) / 2
# Unit direction along DB (from D(1/2, sqrt(3)/2) through B(1, 0)): (1/2, -sqrt(3)/2)
G = model.set_point(
B.x + r_BC * sp.Rational(1, 2),
B.y - r_BC * sp.sqrt(3) / 2,
label="G",
)
H = model.set_point(B.x - r_BC, B.y, ID="H")
# 6. With centre D and distance DG let the circle GKL be described [Post. 3]
model.construct_circle(D, G)
r_DG = 1 + r_BC
# Unit direction along DA (from D(1/2, sqrt(3)/2) through A(0, 0)): (-1/2, -sqrt(3)/2)
L = model.set_point(
D.x - r_DG * sp.Rational(1, 2),
D.y - r_DG * sp.sqrt(3) / 2,
label="L",
)
K = model.set_point(D.x - r_DG, D.y, ID="K")
# Ray endpoints E (beyond L on DA produced) and F (beyond G on DB produced)
r_DE = r_DG + sp.Rational(1, 2)
E = model.set_point(
D.x - r_DE * sp.Rational(1, 2),
D.y - r_DE * sp.sqrt(3) / 2,
label="E",
)
F = model.set_point(
D.x + r_DE * sp.Rational(1, 2),
D.y - r_DE * sp.sqrt(3) / 2,
label="F",
)
# 7. Collinear sections and referenced segments
model.set_section([D, A, L])
model.set_section([D, B, G])
model.set_segment(A, L)
model.set_segment(B, G)
model.set_segment(D, L)
model.set_segment(D, G)
model.set_segment(A, E)
model.set_segment(B, F)
return model
if __name__ == "__main__":
construct_g1()
