G.120 radius to point of contact is perpendicular to tangent — Code¶
If a line touches a circle, the segment joined from the center to the point of contact is perpendicular to the tangent.
"""Programmatic construction for Book III, Proposition 18 (III.18).
If a straight line touch a circle, and a straight line be joined from the
centre to the point of contact, the straight line so joined will be
perpendicular to the tangent.
"""
from __future__ import annotations
import sympy as sp
from geometor.model import Model
def construct_g1() -> Model:
"""Construct the figure for Book III, Proposition 18 using geometor.model."""
model = Model("III.18")
# 1. Circle ABC about centre F, touched by straight line DE at C.
F = model.set_point(0, 0, classes=["given"], ID="F")
C = model.set_point(sp.Rational(4, 5), sp.Rational(-3, 5), classes=["given"], label="C")
B = model.set_point(1, 0, ID="B")
A = model.set_point(sp.Rational(-4, 5), sp.Rational(3, 5), label="A")
circle_ABC = model.construct_circle(F, C)
# 2. Tangent DE through C and radius FC joined.
G = model.set_point(sp.Rational(5, 4), 0, label="G")
D = model.set_point(sp.Rational(23, 16), sp.Rational(1, 4), label="D")
E = model.set_point(sp.Rational(31, 80), sp.Rational(-23, 20), label="E")
line_DE = model.construct_line(D, E)
model.set_segment(D, G)
model.set_segment(G, C)
model.set_segment(C, E)
model.set_segment(D, C)
model.set_segment(G, E)
model.set_segment(D, E)
model.set_section([D, G, C])
model.set_section([G, C, E])
model.set_segment(F, C)
# 3. Reductio: suppose FG is perpendicular to DE, cutting circle ABC at B.
model.set_segment(F, B)
model.set_segment(B, G)
model.set_segment(F, G)
model.set_section([F, B, G])
model.set_polygon([F, C, G])
return model
if __name__ == "__main__":
construct_g1()