1. Type: point()
Definition:
point()
≜
A
::Point
2. Type: line(Point,Point)
Definition:
line(
A
,
B
)
≜
a
::Line
NondegeneracyCondition:¬(is(
A
,
B
))
3. Type: segment(Point,Point)
Definition:
segment(
A
,
B
)
≜
s
::Segment
NondegeneracyCondition:¬(is(
A
,
B
))
4. Type: length(segment(Point,Point))
Definition:
length(segment(
A
,
B
))
≜ [
distance(
A
,
B
)
]
5. Type: halfline(Point,Point)
Definition:
halfline(
A
,
B
)
≜
h
::Halfline
NondegeneracyCondition:¬(is(
A
,
B
))
6. Type: triangle(Point,Point,Point)
Definition:
triangle(
A
,
B
,
C
)
≜
t
::Triangle
NondegeneracyCondition:¬(collinear(
A
,
B
,
C
))
7. Type: triangle(Line,Line,Line)
Definition:
triangle(
l
,
m
,
n
)
≜ [
triangle(intersection(
l
,
m
),intersection(
l
,
n
),intersection(
m
,
n
))
]
NondegeneracyCondition:¬(concurrent(
l
,
m
,
n
))
8. Type: quadrilateral(Point,Point,Point,Point)
Definition:
quadrilateral(
A
,
B
,
C
,
D
)
≜
a
::Quadrilateral
9. Type: angle(Point,Point,Point)
Definition:
angle(
A
,
B
,
C
)
≜
a
::Angle
NondegeneracyCondition:¬(is(
A
,
B
))∧¬(is(
B
,
C
))∧¬(is(
A
,
C
))
10. Type: angle(line(Point,Point),line(Point,Point))
Definition:
angle(line(
A
,
B
),line(
C
,
D
))
≜ [
angle(
A
,intersection(line(
A
,
B
),line(
C
,
D
)),
C
)
where
intersect(line(
A
,
B
),line(
C
,
D
))
]
11. Type: conic(Point,Point,Point,Point,Point)
Definition:
conic(
A
,
B
,
C
,
D
,
E
)
≜
c
::Conic
12. Type: size(angle(Point,Point,Point))
Definition:
size(angle(
A
,
B
,
C
))
≜
s
::Degree
13. Type: circle(Point,Length)
Definition:
circle(
O
,
r
::Length
)
≜
c
::Circle
NondegeneracyCondition:¬(
r
≡
0
::Length
)
14. Type: arc(Point,Point,Point)
Definition:
arc(
A
,
B
,
C
)
≜
a
::Arc
15. Type: perpendicularline(Point,Line)
Definition:
perpendicularline(
A
,
l
)
≜
m
::Line
16. Type: parallelline(Point,Line)
Definition:
parallelline(
A
,
l
)
≜
m
::Line
NondegeneracyCondition:¬(incident(
A
,
l
))
17. Type: is(Point,Point)
Definition:
is(
A
,
B
)
≜
b
::Boolean
18. Type: is(line(Point,Point),Line)
Definition:
is(line(
A
,
B
),
l
)
≜ [
incident(
A
,
l
)∧incident(
B
,
l
)
]
19. Type: is(triangle(Point,Point,Point),triangle(Point,Point,Point))
Definition:
is(triangle(
A
,
B
,
C
),triangle(
D
,
E
,
F
))
≜ [
distance(
A
,
B
)≡distance(
D
,
E
)∧distance(
B
,
C
)≡distance(
E
,
F
)∧distance(
A
,
C
)≡distance(
D
,
F
)
]
20. Type: incident(Point,Circle)
Definition:
incident(
A
,
o
)
≜ [
distance(center(
o
),
A
)≡length(radius(
o
))
]
21. Type: incident(Point,Line)
Definition:
incident(
A
,
l
)
≜
b
::Boolean
22. Type: incident(Point,segment(Point,Point))
Definition:
incident(
A
,segment(
B
,
C
))
≜ [
incident(
A
,line(
B
,
C
))∧differentside(
B
,
C
,
A
)
]
23. Type: incident(Point,line(Point,Point))
Definition:
incident(
A
,line(
B
,
C
))
≜
b
::Boolean
24. Type: incident(Point,halfline(Point,Point))
Definition:
incident(
A
,halfline(
B
,
C
))
≜ [
incident(
A
,line(
B
,
C
))∧sameside(
A
,
C
,
B
)
]
25. Type: incident(Point,perpendicularline(Point,Line))
Definition:
incident(
A
,perpendicularline(
B
,
l
))
≜ [
perpendicular(line(
A
,
B
),
l
)
]
26. Type: incident(Point,parallelline(Point,Line))
Definition:
incident(
A
,parallelline(
B
,
l
))
≜ [
parallel(line(
A
,
B
),
l
)
]
27. Type: incident(Point,bisector(Point,Point,Point))
Definition:
incident(
A
,bisector(
B
,
C
,
D
))
≜ [
distance(
A
,line(
B
,
C
))≡distance(
A
,line(
C
,
D
))
]
28. Type: ratio(GeometricQuantity,GeometricQuantity)
Definition:
ratio(
A
::GeometricQuantity
,
B
::GeometricQuantity
)
≜
a
::AlgebraicQuantity
29. Type: ratio(AlgebraicQuantity,AlgebraicQuantity)
Definition:
ratio(
a
::AlgebraicQuantity
,
b
::AlgebraicQuantity
)
≜
a
::AlgebraicQuantity
30. Type: equal(ratio(GeometricQuantity,GeometricQuantity),ratio(AlgebraicQuantity,AlgebraicQuantity))
Definition:
ratio(
A
::GeometricQuantity
,
B
::GeometricQuantity
)≡ratio(
a
::AlgebraicQuantity
,
b
::AlgebraicQuantity
)
≜ [
b
*
A
≡
a
*
B
]
31. Type: isout(Point,Circle)
Definition:
isout(
A
,
o
)
≜ [
lt(length(radius(
o
)),distance(
A
,center(
o
)))
]
32. Type: isin(Point,Circle)
Definition:
isin(
A
,
o
)
≜ [
lt(distance(
A
,center(
o
)),length(radius(
o
)))
]
33. Type: isin(Point,triangle(Point,Point,Point))
Definition:
isin(
O
,triangle(
A
,
B
,
C
))
≜
b
::Boolean
34. Type: collinear(Point,Point,Point)
Definition:
collinear(
A
,
B
,
C
)
≜ [
incident(
A
,line(
B
,
C
))
]
NondegeneracyCondition:¬(is(
A
,
B
))∧¬(is(
B
,
C
))
35. Type: concurrent(Line,Line,Line)
Definition:
concurrent(
l
,
m
,
n
)
≜ [
incident(intersection(
l
,
m
),
n
)
]
NondegeneracyCondition:¬(is(
l
,
n
))∧¬(is(
m
,
n
))
36. Type: concurrent(Segment,Segment,Segment)
Definition:
concurrent(
l
,
m
,
n
)
≜ [
incident(intersection(
l
,
m
),
n
)
]
NondegeneracyCondition:¬(is(
l
,
n
))∧¬(is(
m
,
n
))
37. Type: perpendicular(line(Point,Point),line(Point,Point))
Definition:
perpendicular(line(
A
,
B
),line(
C
,
D
))
≜
b
::boolean
38. Type: parallel(line(Point,Point),line(Point,Point))
Definition:
parallel(line(
A
,
B
),line(
C
,
D
))
≜
b
::Boolean
39. Type: intersect(Line,Line)
Definition:
intersect(
l
,
m
)
≜ [
¬(parallel(
l
,
m
))
]
40. Type: intersect(Line,Circle)
Definition:
intersect(
l
,
m
)
≜ [
lt(distance(center(
m
),
l
),length(radius(
m
)))
]
41. Type: intersect(Circle,Circle)
Definition:
intersect(
m
,
n
)
≜ [
lt(distance(center(
m
),center(
n
)),length(radius(
m
))+length(radius(
n
)))
]
42. Type: sameside(Point,Point,Line)
Definition:
sameside(
A
,
B
,
l
)
≜
b
::Boolean
43. Type: sameside(Point,Point,Point)
Definition:
sameside(
A
,
B
,
C
)
≜
b
::Boolean
NondegeneracyCondition:collinear(
A
,
B
,
C
)
44. Type: differentside(Point,Point,Line)
Definition:
differentside(
A
,
B
,
l
)
≜
b
::Boolean
45. Type: differentside(Point,Point,Point)
Definition:
differentside(
A
,
B
,
C
)
≜
b
::Boolean
NondegeneracyCondition:collinear(
A
,
B
,
C
)
46. Type: endpoint(segment(Point,Point))
Definition:
endpoint(segment(
A
,
B
))
≜ {[
A
::Point
]; [
B
::Point
]}
47. Type: vertex(triangle(Point,Point,Point))
Definition:
vertex(triangle(
A
,
B
,
C
))
≜ {[
A
::Point
]; [
B
::Point
]; [
C
::Point
]}
48. Type: orthocenter(triangle(Point,Point,Point))
Definition:
orthocenter(triangle(
A
,
B
,
C
))
≜ [
concurrentpoint(altitudelines(triangle(
A
,
B
,
C
)))
]
49. Type: centroid(triangle(Point,Point,Point))
Definition:
centroid(triangle(
A
,
B
,
C
))
≜ [
concurrentpoint(median(
A
,side(
B
,
C
)),median(
B
,side(
A
,
C
)),median(
C
,side(
A
,
B
)))
]
50. Type: barycenter(triangle(Point,Point,Point))
Definition:
barycenter(triangle(
A
,
B
,
C
))
≜ [
centroid(triangle(
A
,
B
,
C
))
]
51. Type: geometriccenter(triangle(Point,Point,Point))
Definition:
geometriccenter(triangle(
A
,
B
,
C
))
≜ [
centroid(triangle(
A
,
B
,
C
))
]
52. Type: incenter(Triangle)
Definition:
incenter(
t
)
≜ [
center(incircle(
t
))
]
53. Type: circumcenter(Triangle)
Definition:
circumcenter(
t
)
≜ [
center(circumcircle(
t
))
]
54. Type: circumradius(Triangle)
Definition:
circumradius(
t
)
≜ [
radius(circumcircle(
t
))
]
55. Type: inradius(Triangle)
Definition:
inradius(
t
)
≜ [
radius(incircle(
t
))
]
56. Type: midperpendicularline(segment(Point,Point))
Definition:
midperpendicularline(segment(
A
,
B
))
≜ [
perpendicularline(midpoint(segment(
A
,
B
)),line(
A
,
B
))
]
57. Type: distance(Point,Point)
Definition:
distance(
A
,
B
)
≜
l
::Length
58. Type: distance(Point,Line)
Definition:
distance(
A
,
l
)
≜ [
distance(
A
,foot(
A
,
l
))
]
59. Type: pointon(Circle)
Definition:
pointon(
o
)
≜
A
::Point
60. Type: pointon(Line)
Definition:
pointon(
l
)
≜
A
::Point
61. Type: midpoint(Point,Point)
Definition:
midpoint(
A
,
B
)
≜ [
midpoint(segment(
A
,
B
))
]
62. Type: midpoint(segment(Point,Point))
Definition:
midpoint(segment(
A
,
B
))
≜
C
::Point
63. Type: circle(Point,Point)
Definition:
circle(
O
,
A
)
≜
o
::Circle
64. Type: circle(Point,Point,Point)
Definition:
circle(
A
,
B
,
C
)
≜
o
::Circle
65. Type: foot(Point,Line)
Definition:
foot(
A
,
l
)
≜ [
intersection(perpendicularline(
A
,
l
),
l
)
]
NondegeneracyCondition:¬(incident(
A
,
l
))
66. Type: intersection(Line,Line)
Definition:
intersection(
l
,
m
)
≜
A
::Point
NondegeneracyCondition:intersect(
l
,
m
)
67. Type: intersection(Segment,Segment)
Definition:
intersection(
l
,
m
)
≜
A
::Point
68. Type: intersection(Line,Circle)
Definition:
intersection(
l
,
o
)
≜
A
::Point
NondegeneracyCondition:intersect(
l
,
o
)
69. Type: intersection(halfline(Point,Point),Circle)
Definition:
intersection(halfline(
A
,
B
),
o
)
≜
C
::Point
NondegeneracyCondition:in(
A
,
o
)
70. Type: intersection(Circle,Circle)
Definition:
intersection(
m
,
n
)
≜
A
::Point
NondegeneracyCondition:intersect(
m
,
n
)
71. Type: tangentpoint(Line,Circle)
Definition:
tangentpoint(
l
,
o
)
≜
A
::Point
NondegeneracyCondition:tangent(
l
,
o
)
72. Type: tangentpoint(Circle,Circle)
Definition:
tangentpoint(
m
,
n
)
≜
A
::Point
NondegeneracyCondition:tangent(
m
,
n
)
73. Type: 2tangentline(Point,Circle)
Definition:
2tangentline(
A
,
o
)
≜
l
::Line
NondegeneracyCondition:out(
A
,
o
)
74. Type: 1tangentline(Point,Circle)
Definition:
1tangentline(
A
,
o
)
≜ [
perpendicularline(
A
,line(
A
,Center(
o
)))
]
NondegeneracyCondition:incident(
A
,
o
)
75. Type: center(Circle)
Definition:
center(
o
)
≜
O
::Point
76. Type: radius(Circle)
Definition:
radius(
o
)
≜ [
segment(center(
o
),pointon(
o
))
]
77. Type: SBintangentcircle(Point,Circle)
Definition:
SBintangentcircle(
A
,
o
)
≜ [
circle(
A
,intersection(halfline(center(
o
),
A
),
o
))
]
NondegeneracyCondition:in(
A
,
o
)∧¬(is(
A
,center(
o
)))
78. Type: BSintangentcircle(Point,Circle)
Definition:
BSintangentcircle(
A
,
o
)
≜ [
circle(
A
,intersection(halfline(
A
,center(
o
)),
o
))
]
NondegeneracyCondition:in(
A
,
o
)∧¬(is(
A
,center(
o
)))
79. Type: extangentcircle(Point,Circle)
Definition:
extangentcircle(
A
,
o
)
≜ [
circle(
A
,intersection(segment(center(
o
),
A
),
o
))
]
NondegeneracyCondition:out(
A
,
o
)
80. Type: tangent(Circle,Circle)
Definition:
tangent(
m
,
n
)
≜ [
(extangent(
m
,
n
)∨intangent(
m
,
n
))
]
81. Type: tangentcircle(Point,Circle)
Definition:
tangentcircle(
A
,
o
)
≜ [
circle(
A
,intersection(line(center(
o
),
A
),
o
))
]
NondegeneracyCondition:¬(is(
A
,center(
o
)))
82. Type: extangent(Circle,Circle)
Definition:
extangent(
m
,
n
)
≜ [
length(radius(
m
))+length(radius(
n
))≡distance(center(
m
),center(
n
))
]
83. Type: intangent(Circle,Circle)
Definition:
intangent(
m
,
n
)
≜ [
(BSintangent(
m
,
n
)∨SBintangent(
m
,
n
))
]
84. Type: BSintangent(Circle,Circle)
Definition:
BSintangent(
m
,
n
)
≜ [
(length(radius(
m
))-length(radius(
n
)))≡distance(center(
m
),center(
n
))
]
85. Type: SBintangent(Circle,Circle)
Definition:
SBintangent(
m
,
n
)
≜ [
(length(radius(
n
))-length(radius(
m
)))≡distance(center(
m
),center(
n
))
]
86. Type: side(Point,Point)
Definition:
side(
A
,
B
)
≜ [
segment(
A
,
B
)
]
87. Type: median(Point,segment(Point,Point))
Definition:
median(
A
,segment(
B
,
C
))
≜ [
segment(
A
,midpoint(segment(
B
,
C
)))
]
NondegeneracyCondition:¬(collinear(
A
,
B
,
C
))
88. Type: medians(triangle(Point,Point,Point))
Definition:
medians(triangle(
A
,
B
,
C
))
≜ {[
median(
A
,segment(
B
,
C
))
]; [
median(
B
,segment(
A
,
C
))
]; [
median(
C
,segment(
A
,
B
))
]}
89. Type: cevian(Point,segment(Point,Point))
Definition:
cevian(
A
,segment(
B
,
C
))
≜ [
segment(
A
,
D
)
where
D
:=pointon(segment(
B
,
C
))
]
NondegeneracyCondition:¬(collinear(
A
,
B
,
C
))
90. Type: altitude(Point,segment(Point,Point))
Definition:
altitude(
A
,segment(
B
,
C
))
≜ [
segment(
A
,foot(
A
,line(
B
,
C
)))
]
NondegeneracyCondition:¬(collinear(
A
,
B
,
C
))
91. Type: altitudes(triangle(Point,Point,Point))
Definition:
altitudes(triangle(
A
,
B
,
C
))
≜ {[
altitude(
A
,segment(
B
,
C
))
]; [
altitude(
B
,segment(
A
,
C
))
]; [
altitude(
C
,segment(
A
,
B
))
]}
92. Type: altitudeline(Point,segment(Point,Point))
Definition:
altitudeline(
A
,segment(
B
,
C
))
≜ [
perpendicularline(
A
,line(
B
,
C
))
]
NondegeneracyCondition:¬(collinear(
A
,
B
,
C
))
93. Type: altitudelines(triangle(Point,Point,Point))
Definition:
altitudelines(triangle(
A
,
B
,
C
))
≜ {[
altitudeline(
A
,segment(
B
,
C
))
]; [
altitudeline(
B
,segment(
A
,
C
))
]; [
altitudeline(
C
,segment(
A
,
B
))
]}
94. Type: bisectorline(Point,Point,Point)
Definition:
bisectorline(
A
,
B
,
C
)
≜ [
line(
B
,
D
)
where
size(angle(
C
,
B
,
D
))≡size(angle(
D
,
B
,
A
))
]
95. Type: bisectorline(Point,Point,Point)
Definition:
bisectorline(
A
,
B
,
C
)
≜
l
::Line
96. Type: bisector(Point,Point,Point)
Definition:
bisector(
B
,
A
,
C
)
≜ [
segment(
A
,intersection(bisectorline(
B
,
A
,
C
),line(
B
,
C
)))
]
NondegeneracyCondition:¬(collinear(
A
,
B
,
C
))
97. Type: bisectors(triangle(Point,Point,Point))
Definition:
bisectors(triangle(
A
,
B
,
C
))
≜ {[
bisector(
A
,
B
,
C
)
]; [
bisector(
B
,
C
,
A
)
]; [
bisector(
C
,
A
,
B
)
]}
98. Type: ninepointcircle(triangle(Point,Point,Point))
Definition:
ninepointcircle(triangle(
A
,
B
,
C
))
≜ [
circle(midpoint(
A
,
B
),midpoint(
A
,
C
),midpoint(
B
,
C
))
]
99. Type: Eulerpoint(Point,Triangle)
Definition:
Eulerpoint(
A
,
t
)
≜ [
midpoint(segment(
A
,orthocenter(
t
)))
]
100. Type: Eulerline(Triangle)
Definition:
Eulerline(
a
)
≜ [
collinearline(orthocenter(
a
),centroid(
a
),circumcenter(
a
))
]
101. Type: collinearline(Point,Point,Point)
Definition:
collinearline(
A
,
B
,
C
)
≜ [
line(choosediff(
A
B
C
,
2
::Number
))
]
NondegeneracyCondition:collinear(
A
,
B
,
C
)
102. Type: concurrentpoint(Line,Line,Line)
Definition:
concurrentpoint(
l
,
m
,
n
)
≜ [
intersection(choosediff(
l
m
n
,
2
::Number
))
]
NondegeneracyCondition:concurrent(
l
,
m
,
n
)
103. Type: concurrentpoint(segment(Point,Point),segment(Point,Point),segment(Point,Point))
Definition:
concurrentpoint(segment(
A
,
B
),segment(
C
,
D
),segment(
E
,
F
))
≜ [
concurrentpoint(line(
A
,
B
),line(
C
,
D
),line(
E
,
F
))
]
104. Type: circumcircle(triangle(Point,Point,Point))
Definition:
circumcircle(triangle(
A
,
B
,
C
))
≜ [
circle(
A
,
B
,
C
)
]
105. Type: incircle(triangle(Point,Point,Point))
Definition:
incircle(triangle(
A
,
B
,
C
))
≜ [
circle(
O
,foot(
O
,line(
A
,
B
)))
where
O
:=intersection(bisectorline(
A
,
B
,
C
),bisectorline(
B
,
C
,
A
))
]
106. Type: excircle(Point,side(Point,Point))
Definition:
excircle(
C
,side(
A
,
B
))
≜ [
circle(
O
,foot(
O
,line(
A
,
B
)))
where
O
:=intersection(bisectorline(
C
,
B
,
A
),bisectorline(
B
,
A
,
C
))
]
107. Type: escenter(Point,side(Point,Point))
Definition:
escenter(
C
,side(
A
,
B
))
≜ [
intersection(bisectorline(
C
,
B
,
A
),bisectorline(
B
,
A
,
C
))
]
108. Type: escenter(triangle(Point,Point,Point))
Definition:
escenter(triangle(
A
,
B
,
C
))
≜ {[
escenter(
A
,side(
B
,
C
))
]; [
escenter(
B
,side(
A
,
C
))
]; [
escenter(
C
,side(
A
,
B
))
]}
109. Type: excircle(triangle(Point,Point,Point))
Definition:
excircle(triangle(
A
,
B
,
C
))
≜ {[
excircle(
A
,side(
B
,
C
))
]; [
excircle(
B
,side(
A
,
C
))
]; [
excircle(
C
,side(
A
,
B
))
]}
110. Type: tritangentcircle(triangle(Point,Point,Point))
Definition:
tritangentcircle(triangle(
A
,
B
,
C
))
≜ {[
incircle(triangle(
A
,
B
,
C
))
]; [
excircle(
A
,side(
B
,
C
))
]; [
excircle(
B
,side(
A
,
C
))
]; [
excircle(
C
,side(
A
,
B
))
]}
111. Type: equilateraltriangle(Point,Point,Point)
Definition:
equilateraltriangle(
A
,
B
,
C
)
≜ [
triangle(
A
,
B
,rotate(
A
,
B
,
pi/3
::Degree
))
]
112. Type: externalequilateraltriangle(segment(Point,Point),Point)
Definition:
externalequilateraltriangle(segment(
A
,
B
),
C
)
≜ [
triangle(
A
,
B
,rotate(
A
,
B
,
pi/3
::Degree
))
]
NondegeneracyCondition:¬(incidentLine(
C
,line(
A
,
B
)))
113. Type: Napoleontriangle(triangle(Point,Point,Point))
Definition:
Napoleontriangle(triangle(
A
,
B
,
C
))
≜ [
triangle(barycenter(externalequilateraltriangle(side(
B
,
C
),
A
)),barycenter(externalequilateraltriangle(side(
C
,
A
),
B
)),barycenter(externalequilateraltriangle(side(
A
,
B
),
C
)))
]
114. Type: Pappus(Point,Point,Point,Point,Point,Point,Point,Point,Point)
Definition:
Pappus(
A
,
B
,
C
,
D
,
E
,
F
,
P
,
Q
,
R
)
≜ [
C
:=pointon(line(
A
,
B
));
F
:=pointon(line(
D
,
E
));
P
:=intersection(line(
A
,
E
),line(
D
,
B
));
Q
:=intersection(line(
A
,
F
),line(
D
,
C
));
R
:=intersection(line(
B
,
F
),line(
E
,
C
));
]
115. Type: concyclic(Point,Point,Point,Point,Point,Point)
Definition:
concyclic(
A
,
B
,
C
,
D
,
E
,
F
)
≜ [
O
:=center(circle(
A
,
B
,
C
));
D
:=pointon(circle(
O
,
A
));
E
:=pointon(circle(
O
,
A
));
F
:=pointon(circle(
O
,
A
));
]
116. Type: concyclic(Point,Point,Point,Point)
Definition:
concyclic(
A
,
B
,
C
,
D
)
≜ [
incident(
D
,circle(
A
,
B
,
C
))
]
117. Type: completequadrilateral(Point,Point,Point,Point,Point,Point)
Definition:
completequadrilateral(
A
,
B
,
C
,
D
,
E
,
F
)
≜ [
E
:=intersection(line(
A
,
B
),line(
C
,
D
));
F
:=intersection(line(
A
,
C
),line(
B
,
D
));
]
118. Type: diagonal(completequadrilateral(Point,Point,Point,Point,Point,Point))
Definition:
diagonal(completequadrilateral(
A
,
B
,
C
,
D
,
E
,
F
))
≜ {[
segment(
A
,
D
)
]; [
segment(
B
,
C
)
]; [
segment(
E
,
F
)
]}
119. Type: Gaussline(Point,Point,Point,Point)
Definition:
Gaussline(
A
,
B
,
C
,
D
)
≜ [
collinearline(midpoint(diagonal(completequadrilateral(
A
,
B
,
C
,
D
,
E
,
F
))))
]
120. Type: Simsonline(Point,triangle(Point,Point,Point))
Definition:
Simsonline(
D
,triangle(
A
,
B
,
C
))
≜ [
collinearline(foot(
D
,side(
A
,
B
)),foot(
D
,side(
A
,
C
)),foot(
D
,side(
B
,
C
)))
]
121. Type: mediator(segment(Point,Point))
Definition:
mediator(segment(
A
,
B
))
≜ [
midperpendicularline(segment(
A
,
B
))
]
122. Type: orthictriangle(triangle(Point,Point,Point))
Definition:
orthictriangle(triangle(
A
,
B
,
C
))
≜ [
triangle(foot(
A
,line(
B
,
C
)),foot(
B
,line(
A
,
C
)),foot(
C
,line(
A
,
B
)))
]
123. Type: homothetic(triangle(Point,Point,Point),triangle(Point,Point,Point))
Definition:
homothetic(triangle(
A
,
B
,
C
),triangle(
A
,
B
,
C
))
≜ [
parallel(line(
A
,
B
),line(
D
,
E
))∧parallel(line(
A
,
C
),line(
D
,
F
))∧parallel(line(
B
,
C
),line(
E
,
F
))
]
124. Type: homotheticcenter(triangle(Point,Point,Point),triangle(Point,Point,Point))
Definition:
homotheticcenter(triangle(
A
,
B
,
C
),triangle(
D
,
E
,
F
))
≜ [
concurrentpoint(line(
A
,
D
),line(
B
,
E
),line(
C
,
F
))
]
125. Type: tangentialtriangle(triangle(Point,Point,Point))
Definition:
tangentialtriangle(triangle(
A
,
B
,
C
))
≜ [
triangle(perpendicularline(
A
,line(
O
,
A
)),perpendicularline(
B
,line(
O
,
B
)),perpendicularline(
C
,line(
O
,
C
)))
where
O
:=circumcenter(triangle(
A
,
B
,
C
))
]
126. Type: antiparallel(Line,Line,Line)
Definition:
antiparallel(
l
,
m
,
n
)
≜ [
size(angle(
l
,
n
))≡size(angle(
n
,
m
))
]
127. Type: orthorcentricgroup(Point,triangle(Point,Point,Point))
Definition:
orthorcentricgroup(
H
,triangle(
A
,
B
,
C
))
≜ [
H
:=orthocenter(triangle(
A
,
B
,
C
));
]
128. Type: circumdiameterline(Point,side(Point,Point))
Definition:
circumdiameterline(
A
,side(
B
,
C
))
≜ [
line(
A
,circumcenter(triangle(
A
,
B
,
C
)))
]
129. Type: symmetry(Point,line(Point,Point))
Definition:
symmetry(
A
,line(
B
,
C
))
≜ [
symmetry(
A
,foot(
A
,line(
B
,
C
)))
]
130. Type: symmetry(Point,Point)
Definition:
symmetry(
A
,
B
)
≜
C
::Point
131. Type: anticomplementarytriangle(triangle(Point,Point,Point))
Definition:
anticomplementarytriangle(triangle(
A
,
B
,
C
))
≜ [
triangle(
D
,
E
,
F
)
where
is(
A
,midpoint(segment(
D
,
E
)))∧is(
B
,midpoint(segment(
E
,
F
)))∧is(
C
,midpoint(segment(
D
,
F
)))
]
132. Type: anticenter(cyclicquadrilateral(Point,Point,Point,Point))
Definition:
anticenter(cyclicquadrilateral(
A
,
B
,
C
,
D
))
≜ [
concurrentpoint(maltitude(quadrilateral(
A
,
B
,
C
,
D
)))
]
133. Type: maltitude(quadrilateral(Point,Point,Point,Point))
Definition:
maltitude(quadrilateral(
A
,
B
,
C
,
D
))
≜ {[
perpendicularline(midpoint(
A
,
B
),line(
C
,
D
))
]; [
perpendicularline(midpoint(
B
,
C
),line(
A
,
D
))
]; [
perpendicularline(midpoint(
C
,
D
),line(
A
,
B
))
]; [
perpendicularline(midpoint(
A
,
D
),line(
B
,
C
))
]}
134. Type: cyclicquadrilateral(Point,Point,Point,Point)
Definition:
cyclicquadrilateral(
A
,
B
,
C
,
D
)
≜ [
quadrilateral(
A
,
B
,
C
,pointon(circle(
A
,
B
,
C
)))
]
135. Type: orthodiagonalquadrilateral(Point,Point,Point,Point)
Definition:
orthodiagonalquadrilateral(
A
,
B
,
C
,
D
)
≜ [
quadrilateral(
A
,
B
,
C
,pointon(perpendicularline(
B
,line(
A
,
C
))))
]
136. Type: Miquelcircle(Point,Point,Point,side(Point,Point))
Definition:
Miquelcircle(
A
,
D
,
E
,side(
B
,
C
))
≜ [
circle(
A
,pointon(line(
A
,
B
)),pointon(line(
A
,
C
)))
]
137. Type: contacttriangle(triangle(Point,Point,Point))
Definition:
contacttriangle(triangle(
A
,
B
,
C
))
≜ [
triangle(foot(
O
,line(
A
,
B
)),foot(
O
,line(
A
,
C
)),foot(
O
,line(
B
,
C
)))
where
O
:=incenter(triangle(
A
,
B
,
C
))
]
138. Type: rotate(Point,Point,Degree)
Definition:
rotate(
A
,
B
,
d
::Degree
)
≜
C
::Point
NondegeneracyCondition:¬(is(
A
,
B
))
139. Type: parallelogram(Point,Point,Point,Point)
Definition:
parallelogram(
A
,
B
,
C
,
D
)
≜ [
quadrilateral(
A
,
B
,
C
,
D
)
where
D
:=intersection(parallelline(
A
,line(
B
,
C
)),parallelline(
C
,line(
A
,
B
)))
]