def is_isosceles(graph, this_triangle):
is_isosceles_result = False
vertex_point = ''
aside_points = ()
""" every vertex """
for point_vertex in this_triangle:
point_aside = tuple_del(this_triangle, point_vertex)
if (graph.query(
equal(segment(point_aside[0], point_vertex),
segment(point_aside[1], point_vertex)),
'segment')) or (graph.query(
equal(angle(point_vertex, point_aside[0], point_aside[1]),
angle(point_vertex, point_aside[1], point_aside[0])),
'angle')):
is_isosceles_result = True
vertex_point = point_vertex
aside_points = point_aside
return is_isosceles_result, vertex_point, aside_points
""" return judge result """
return is_isosceles_result, vertex_point, aside_points
def quadrilateral_theorem_transform(graph):
triangles_list = graph.triangles_list
for this_triangle in triangles_list:
out_points = tuple_except_tuple(graph.points.keys(),
this_triangle)
for point_vertex in this_triangle:
point_aside = tuple_del(this_triangle, point_vertex)
for point_out in out_points:
""" coexist with a circle """
if graph.query(
equal(
sum_units(
angle(point_out, point_vertex,
point_aside[0]),
angle(point_out, point_vertex,
point_aside[1])),
angle(point_aside[0], point_vertex,
point_aside[1])), 'angle'):
if graph.query(equal(sum_units(angle(point_aside[0], point_vertex, point_aside[1]),
angle(point_aside[0], point_out, point_aside[1])),
degree(180)), 'angle') or \
graph.query(equal(angle(point_out, point_vertex, point_aside[1]),
angle(point_out, point_aside[0], point_aside[1])), 'angle') or \
graph.query(equal(angle(point_out, point_vertex, point_aside[0]),
angle(point_out, point_aside[1], point_aside[0])), 'angle'):
graph.add_equal(
equal(
angle(point_out, point_vertex,
point_aside[1]),
angle(point_out, point_aside[0],
point_aside[1])), 'angle')
graph.add_equal(
equal(
angle(point_out, point_vertex,
point_aside[0]),
angle(point_out, point_aside[1],
point_aside[0])), 'angle')
graph.add_equal(
equal(
angle(point_vertex, point_out,
point_aside[1]),
angle(point_vertex, point_aside[0],
point_aside[1])), 'angle')
graph.add_equal(
equal(
angle(point_vertex, point_out,
point_aside[0]),
angle(point_vertex, point_aside[1],
point_aside[0])), 'angle')
def is_congruent(graph, triangle_1, triangle_2):
is_congruent_result = False
three_points_1 = ()
three_points_2 = ()
""" every double vertex """
for point_vertex_1 in triangle_1:
point_aside_1 = tuple_del(triangle_1, point_vertex_1)
for point_vertex_2 in triangle_2:
point_aside_2 = tuple_del(triangle_2, point_vertex_2)
""" A or S """
if graph.query(
equal(
angle(point_aside_1[0], point_vertex_1,
point_aside_1[1]),
angle(point_aside_2[0], point_vertex_2,
point_aside_2[1])), 'angle'):
for point_aside_1_1 in point_aside_1:
point_aside_1_2 = tuple_del(point_aside_1,
point_aside_1_1)[0]
point_aside_2_1 = point_aside_2[0]
point_aside_2_2 = point_aside_2[1]
""" AS """
if graph.query(
equal(segment(point_aside_1_1, point_vertex_1),
segment(point_aside_2_1, point_vertex_2)),
'segment'):
""" ASS """
if graph.query(
equal(segment(point_aside_1_2, point_vertex_1),
segment(point_aside_2_2,
point_vertex_2)), 'segment'):
is_congruent_result = True
three_points_1 = (point_aside_1_1, point_vertex_1,
point_aside_1_2)
three_points_2 = (point_aside_2_1, point_vertex_2,
point_aside_2_2)
return is_congruent_result, three_points_1, three_points_2
elif graph.query(
equal(segment(point_aside_1[0], point_aside_1[1]),
segment(point_aside_2[0], point_aside_2[1])),
'segment'):
for point_aside_1_1 in point_aside_1:
point_aside_1_2 = tuple_del(point_aside_1,
point_aside_1_1)[0]
point_aside_2_1 = point_aside_2[0]
point_aside_2_2 = point_aside_2[1]
""" SS or SA"""
if graph.query(
equal(segment(point_aside_1_1, point_vertex_1),
segment(point_aside_2_1, point_vertex_2)),
'segment'):
""" SSS """
if graph.query(
equal(segment(point_aside_1_2, point_vertex_1),
segment(point_aside_2_2,
point_vertex_2)), 'segment'):
is_congruent_result = True
three_points_1 = (point_aside_1_1, point_vertex_1,
point_aside_1_2)
three_points_2 = (point_aside_2_1, point_vertex_2,
point_aside_2_2)
return is_congruent_result, three_points_1, three_points_2
elif graph.query(
equal(
angle(point_vertex_1, point_aside_1_1,
point_aside_1_2),
angle(point_vertex_2, point_aside_2_1,
point_aside_2_2)), 'angle'):
""" SAA """
if graph.query(
equal(
angle(point_vertex_1, point_aside_1_2,
point_aside_1_1),
angle(point_vertex_2, point_aside_2_2,
point_aside_2_1)), 'angle'):
is_congruent_result = True
three_points_1 = (point_aside_1_1, point_vertex_1,
point_aside_1_2)
three_points_2 = (point_aside_2_1, point_vertex_2,
point_aside_2_2)
return is_congruent_result, three_points_1, three_points_2
""" return judge result """
return is_congruent_result, three_points_1, three_points_2
def space_and_collinear_transform(graph):
""" find collinear """
three_points_list = list(combinations(graph.points.keys(), 3))
triangles_list = three_points_list.copy()
collinear_list = []
for this_three_point in three_points_list:
yes_collinear, vertex_point, aside_points = is_collinear(
graph, this_three_point)
if yes_collinear:
triangles_list.remove(this_three_point)
collinear_list.append(
(aside_points[0], vertex_point, aside_points[1]))
graph.triangles_list = triangles_list
graph.collinear_list = collinear_list
""" collinear transform """
for this_collinear in collinear_list:
graph.add_equal(
equal(
sum_units(
segment(this_collinear[0], this_collinear[1]),
segment(this_collinear[2], this_collinear[1])),
segment(this_collinear[0], this_collinear[2])),
'segment')
out_points = tuple_except_tuple(graph.points.keys(),
this_collinear)
for point_out in out_points:
if not is_collinear(
graph,
(point_out, this_collinear[0], this_collinear[1]))[0]:
graph.add_equal(
equal(
sum_units(
angle(point_out, this_collinear[1],
this_collinear[0]),
angle(point_out, this_collinear[1],
this_collinear[2])), degree(180)),
'angle')
graph.add_equal(
equal(
angle(point_out, this_collinear[0],
this_collinear[1]),
angle(point_out, this_collinear[0],
this_collinear[2])), 'angle')
graph.add_equal(
equal(
angle(point_out, this_collinear[2],
this_collinear[1]),
angle(point_out, this_collinear[2],
this_collinear[0])), 'angle')
""" not collinear transform """
for this_triangle in triangles_list:
out_points = tuple_except_tuple(graph.points.keys(),
this_triangle)
for point_vertex in this_triangle:
point_aside = tuple_del(this_triangle, point_vertex)
for point_out in out_points:
degree_1 = cal_an_angle_degree(graph, point_out,
point_vertex,
point_aside[0])
degree_2 = cal_an_angle_degree(graph, point_out,
point_vertex,
point_aside[1])
degree_sum = cal_an_angle_degree(
graph, point_aside[0], point_vertex,
point_aside[1])
if degree_1 > 5.0 and degree_2 > 5.0 and abs(
degree_sum - degree_1 - degree_2) < 2.0:
graph.add_equal(
equal(
sum_units(
angle(point_out, point_vertex,
point_aside[0]),
angle(point_out, point_vertex,
point_aside[1])),
angle(point_aside[0], point_vertex,
point_aside[1])), 'angle')
def triangle_theorem_transform(graph):
triangles_list = graph.triangles_list
triangles_num = len(triangles_list)
for no_1 in range(triangles_num):
this_triangle = triangles_list[no_1]
""" interior angle summation 180° """
graph.add_equal(
equal(
sum_units(
angle(this_triangle[1], this_triangle[0],
this_triangle[2]),
angle(this_triangle[0], this_triangle[1],
this_triangle[2]),
angle(this_triangle[0], this_triangle[2],
this_triangle[1])), degree(180)), 'angle')
""" isosceles """
yes_isosceles, vertex_point, aside_points = is_isosceles(
graph, this_triangle)
if yes_isosceles:
graph.add_equal(
equal(segment(vertex_point, aside_points[0]),
segment(vertex_point, aside_points[1])),
'segment')
graph.add_equal(
equal(
angle(vertex_point, aside_points[0],
aside_points[1]),
angle(vertex_point, aside_points[1],
aside_points[0])), 'angle')
""" congruent and similar """
for no_2 in range(no_1 + 1, triangles_num):
another_triangle = triangles_list[no_2]
""" congruent """
yes_congruent, three_points_1, three_points_2 = is_congruent(
graph, this_triangle, another_triangle)
if yes_congruent:
graph.add_equal(
equal(
segment(three_points_1[0], three_points_1[1]),
segment(three_points_2[0], three_points_2[1])),
'segment')
graph.add_equal(
equal(
segment(three_points_1[0], three_points_1[2]),
segment(three_points_2[0], three_points_2[2])),
'segment')
graph.add_equal(
equal(
segment(three_points_1[1], three_points_1[2]),
segment(three_points_2[1], three_points_2[2])),
'segment')
graph.add_equal(
equal(
angle(three_points_1[1], three_points_1[0],
three_points_1[2]),
angle(three_points_2[1], three_points_2[0],
three_points_2[2])), 'angle')
graph.add_equal(
equal(
angle(three_points_1[0], three_points_1[1],
three_points_1[2]),
angle(three_points_2[0], three_points_2[1],
three_points_2[2])), 'angle')
graph.add_equal(
equal(
angle(three_points_1[0], three_points_1[2],
three_points_1[1]),
angle(three_points_2[0], three_points_2[2],
three_points_2[1])), 'angle')
""" similar """
yes_similar, three_points_1, three_points_2 = is_similar(
graph, this_triangle, another_triangle)
if yes_similar:
graph.add_equal(
equal(
multiply_units(
segment(three_points_1[0],
three_points_1[1]),
segment(three_points_2[1],
three_points_2[2])),
multiply_units(
segment(three_points_1[1],
three_points_1[2]),
segment(three_points_2[0],
three_points_2[1]))), 'segment')
graph.add_equal(
equal(
multiply_units(
segment(three_points_1[0],
three_points_1[1]),
segment(three_points_2[0],
three_points_2[2])),
multiply_units(
segment(three_points_1[0],
three_points_1[2]),
segment(three_points_2[0],
three_points_2[1]))), 'segment')
graph.add_equal(
equal(
multiply_units(
segment(three_points_1[0],
three_points_1[2]),
segment(three_points_2[1],
three_points_2[2])),
multiply_units(
segment(three_points_1[1],
three_points_1[2]),
segment(three_points_2[0],
three_points_2[2]))), 'segment')
graph.add_equal(
equal(
angle(three_points_1[1], three_points_1[0],
three_points_1[2]),
angle(three_points_2[1], three_points_2[0],
three_points_2[2])), 'angle')
graph.add_equal(
equal(
angle(three_points_1[0], three_points_1[1],
three_points_1[2]),
angle(three_points_2[0], three_points_2[1],
three_points_2[2])), 'angle')
graph.add_equal(
equal(
angle(three_points_1[0], three_points_1[2],
three_points_1[1]),
angle(three_points_2[0], three_points_2[2],
three_points_2[1])), 'angle')
def complex_units_equal_transform(graph, equals_list, equal_type):
for each_equal in equals_list:
for each_complex_unit in each_equal:
if is_complex_type(each_complex_unit):
complex_type = get_unit_type(each_complex_unit)
""" replace sub unit """
""" AB+BM=MN, BM=BC ---> AB+BM=AB+BC=MN """
""" AB*BM=MN*ML, BM=BC ---> AB*BM=AB*BC=MN*ML """
""" AB*AB=MN*ML, AB=BC ---> AB*AB=AB*BC=BC*BC=MN*ML """
for each_sub_unit in get_complexunit_set(
each_complex_unit):
for each_another_equal in equals_list:
if each_sub_unit in each_another_equal:
for each_another_sub_unit in get_equal_except_unit(
each_another_equal, each_sub_unit):
new_complex_unit = complex_units(
complex_type,
get_complexunit_except_unit(
each_complex_unit,
each_sub_unit),
each_another_sub_unit)
if equal_type == 'angle' and get_unit_len(
new_complex_unit
) > deduce_config[
'angle_complex_max_len']:
continue
graph.add_equal(
equal(new_complex_unit,
each_complex_unit),
equal_type)
""" eliminate sub unit """
""" AB+BM=AB+BC ---> BM=BC ; AB+BM+MN=AB+BC ---> BM+MN=BC """
""" AB*BM=AB*BC ---> BM=BC ; AB*BM*MN=AB*BC*DE ---> BM*MN=BC*DE """
""" AB*AB=AB*BC ---> AB=BC ; AB*AB*MN=AB*BC*DE ---> AB*MN=BC*DE """
for each_another_complex_unit in get_equal_except_unit(
each_equal, each_complex_unit):
if get_unit_type(
each_another_complex_unit) == complex_type:
if get_unit_len(
get_complexunit_inner_complexunit(
each_complex_unit,
each_another_complex_unit)) >= 1:
graph.add_equal(
equal(
get_complexunit_except_unit(
each_complex_unit,
each_another_complex_unit),
get_complexunit_except_unit(
each_another_complex_unit,
each_complex_unit)),
equal_type)
""" multiply allocate """
""" (AE+EB)*AC=AB*AC ---> (AE+EB)*AC=AE*AC+EB*AC=AB*AC """
""" (AE+EM+MB)*AC=AB*AC ---> (AE+EM+MB)*AC=AE*AC+EM*AC+MB*AC=AB*AC """
""" (AE+AE+MB)*AC=AB*AC ---> (AE+AE+MB)*AC=AE*AC+AE*AC+MB*AC=AB*AC """
if complex_type == 'multiply':
for each_sub_unit in get_complexunit_set(
each_complex_unit):
if get_unit_type(each_sub_unit) == 'sum':
excepted_unit = get_complexunit_except_unit(
each_complex_unit, each_sub_unit)
allocated_complex_unit = sum_units()
for each_sub_sub_unit in get_complexunit_set(
each_sub_unit):
allocated_complex_unit = sum_units(
allocated_complex_unit,
multiply_units(
each_sub_sub_unit,
excepted_unit))
graph.add_equal(
equal(allocated_complex_unit,
each_complex_unit), equal_type)
""" multiply associate """
""" AE*AC+EB*AC=AB*AC ---> AE*AC+EB*AC=(AE+EB)*AC=AB*AC """
""" AE*AC+EM*AC+MB*AC=AB*AC ---> AE*AC+EM*AC+MB*AC=(AE+EM)*AC+MB*AC=AB*AC """
""" (AE+EM)*AC+MB*AC=AB*AC ---> (AE+EM)*AC+MB*AC=(AE+EM+MB)*AC=AB*AC """
if complex_type == 'sum':
for each_sub_unit in get_complexunit_set(
each_complex_unit):
if get_unit_type(each_sub_unit) == 'multiply':
excepted_unit = get_complexunit_except_unit(
each_complex_unit, each_sub_unit)
if get_unit_type(excepted_unit) == 'sum':
for each_another_sub_unit in get_complexunit_set(
excepted_unit):
if get_unit_type(
each_another_sub_unit
) == 'multiply':
inner_unit = get_complexunit_inner_complexunit(
each_sub_unit,
each_another_sub_unit)
excepted_2_unit = get_complexunit_except_unit(
excepted_unit,
each_another_sub_unit)
for inner_sub_unit in get_unit_flatten_set(
inner_unit,
'multiply'):
associated_complex_unit = sum_units(
multiply_units(
inner_sub_unit,
sum_units(
get_complexunit_except_unit(
each_sub_unit,
inner_sub_unit
),
get_complexunit_except_unit(
each_another_sub_unit,
inner_sub_unit
))),
excepted_2_unit)
graph.add_equal(
equal(
associated_complex_unit,
each_complex_unit),
equal_type)
elif get_unit_type(
excepted_unit) == 'multiply':
inner_unit = get_complexunit_inner_complexunit(
each_sub_unit, excepted_unit)
for inner_sub_unit in get_unit_flatten_set(
inner_unit, 'multiply'):
associated_complex_unit = multiply_units(
inner_sub_unit,
sum_units(
get_complexunit_except_unit(
each_sub_unit,
inner_sub_unit),
get_complexunit_except_unit(
excepted_unit,
inner_sub_unit)))
graph.add_equal(
equal(associated_complex_unit,
each_complex_unit),
equal_type)