Polygons

class sympy.geometry.polygon.Polygon[source]

A two-dimensional polygon.

A simple polygon in space. Can be constructed from a sequence of points or from a center, radius, number of sides and rotation angle.

Parameters :

vertices : sequence of Points

Raises :

GeometryError :

If all parameters are not Points.

If the Polygon has intersecting sides.

Notes

Polygons are treated as closed paths rather than 2D areas so some calculations can be be negative or positive (e.g., area) based on the orientation of the points.

Any consecutive identical points are reduced to a single point and any points collinear and between two points will be removed unless they are needed to define an explicit intersection (see examples).

A Triangle, Segment or Point will be returned when there are 3 or fewer points provided.

Examples

>>> from sympy import Point, Polygon, pi
>>> p1, p2, p3, p4, p5 = [(0, 0), (1, 0), (5, 1), (0, 1), (3, 0)]
>>> Polygon(p1, p2, p3, p4)
Polygon(Point(0, 0), Point(1, 0), Point(5, 1), Point(0, 1))
>>> Polygon(p1, p2)
Segment(Point(0, 0), Point(1, 0))
>>> Polygon(p1, p2, p5)
Segment(Point(0, 0), Point(3, 0))

While the sides of a polygon are not allowed to cross implicitly, they can do so explicitly. For example, a polygon shaped like a Z with the top left connecting to the bottom right of the Z must have the point in the middle of the Z explicitly given:

>>> mid = Point(1, 1)
>>> Polygon((0, 2), (2, 2), mid, (0, 0), (2, 0), mid).area
0
>>> Polygon((0, 2), (2, 2), mid, (2, 0), (0, 0), mid).area
-2

When the the keyword \(n\) is used to define the number of sides of the Polygon then a RegularPolygon is created and the other arguments are interpreted as center, radius and rotation. The unrotated RegularPolygon will always have a vertex at Point(r, 0) where \(r\) is the radius of the circle that circumscribes the RegularPolygon. Its method \(spin\) can be used to increment that angle.

>>> p = Polygon((0,0), 1, n=3)
>>> p
RegularPolygon(Point(0, 0), 1, 3, 0)
>>> p.vertices[0]
Point(1, 0)
>>> p.args[0]
Point(0, 0)
>>> p.spin(pi/2)
>>> p.vertices[0]
Point(0, 1)

Attributes

area  
angles  
perimeter  
vertices  
centroid  
sides  
angles[source]

The internal angle at each vertex.

Returns :

angles : dict

A dictionary where each key is a vertex and each value is the internal angle at that vertex. The vertices are represented as Points.

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.angles[p1]
pi/2
>>> poly.angles[p2]
acos(-4*sqrt(17)/17)
arbitrary_point(parameter='t')[source]

A parameterized point on the polygon.

The parameter, varying from 0 to 1, assigns points to the position on the perimeter that is that fraction of the total perimeter. So the point evaluated at t=1/2 would return the point from the first vertex that is 1/2 way around the polygon.

Parameters :

parameter : str, optional

Default value is ‘t’.

Returns :

arbitrary_point : Point

Raises :

ValueError :

When \(parameter\) already appears in the Polygon’s definition.

Examples

>>> from sympy import Polygon, S, Symbol
>>> t = Symbol('t', real=True)
>>> tri = Polygon((0, 0), (1, 0), (1, 1))
>>> p = tri.arbitrary_point('t')
>>> perimeter = tri.perimeter
>>> s1, s2 = [s.length for s in tri.sides[:2]]
>>> p.subs(t, (s1 + s2/2)/perimeter)
Point(1, 1/2)
area[source]

The area of the polygon.

Notes

The area calculation can be positive or negative based on the orientation of the points.

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.area
3
centroid[source]

The centroid of the polygon.

Returns :centroid : Point

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.centroid
Point(31/18, 11/18)
distance(o)[source]

Returns the shortest distance between self and o.

If o is a point, then self does not need to be convex. If o is another polygon self and o must be complex.

Examples

>>> from sympy import Point, Polygon, RegularPolygon
>>> p1, p2 = map(Point, [(0, 0), (7, 5)])
>>> poly = Polygon(*RegularPolygon(p1, 1, 3).vertices)
>>> poly.distance(p2)
sqrt(61)
encloses_point(p)[source]

Return True if p is enclosed by (is inside of) self.

Parameters :p : Point
Returns :encloses_point : True, False or None

Notes

Being on the border of self is considered False.

References

[1] http://www.ariel.com.au/a/python-point-int-poly.html

Examples

>>> from sympy import Polygon, Point
>>> from sympy.abc import t
>>> p = Polygon((0, 0), (4, 0), (4, 4))
>>> p.encloses_point(Point(2, 1))
True
>>> p.encloses_point(Point(2, 2))
False
>>> p.encloses_point(Point(5, 5))
False
intersection(o)[source]

The intersection of two polygons.

The intersection may be empty and can contain individual Points and complete Line Segments.

Parameters :

other: Polygon :

Returns :

intersection : list

The list of Segments and Points

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly1 = Polygon(p1, p2, p3, p4)
>>> p5, p6, p7 = map(Point, [(3, 2), (1, -1), (0, 2)])
>>> poly2 = Polygon(p5, p6, p7)
>>> poly1.intersection(poly2)
[Point(2/3, 0), Point(9/5, 1/5), Point(7/3, 1), Point(1/3, 1)]
is_convex()[source]

Is the polygon convex?

A polygon is convex if all its interior angles are less than 180 degrees.

Returns :

is_convex : boolean

True if this polygon is convex, False otherwise.

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.is_convex()
True
perimeter[source]

The perimeter of the polygon.

Returns :perimeter : number or Basic instance

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.perimeter
sqrt(17) + 7
plot_interval(parameter='t')[source]

The plot interval for the default geometric plot of the polygon.

Parameters :

parameter : str, optional

Default value is ‘t’.

Returns :

plot_interval : list (plot interval)

[parameter, lower_bound, upper_bound]

Examples

>>> from sympy import Polygon
>>> p = Polygon((0, 0), (1, 0), (1, 1))
>>> p.plot_interval()
[t, 0, 1]
sides[source]

The line segments that form the sides of the polygon.

Returns :

sides : list of sides

Each side is a Segment.

Notes

The Segments that represent the sides are an undirected line segment so cannot be used to tell the orientation of the polygon.

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.sides
[Segment(Point(0, 0), Point(1, 0)),
Segment(Point(1, 0), Point(5, 1)),
Segment(Point(0, 1), Point(5, 1)), Segment(Point(0, 0), Point(0, 1))]
vertices[source]

The vertices of the polygon.

Returns :vertices : tuple of Points

Notes

When iterating over the vertices, it is more efficient to index self rather than to request the vertices and index them. Only use the vertices when you want to process all of them at once. This is even more important with RegularPolygons that calculate each vertex.

Examples

>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.vertices
(Point(0, 0), Point(1, 0), Point(5, 1), Point(0, 1))
>>> poly.args[0]
Point(0, 0)
class sympy.geometry.polygon.RegularPolygon[source]

A regular polygon.

Such a polygon has all internal angles equal and all sides the same length.

Parameters :

center : Point

radius : number or Basic instance

The distance from the center to a vertex

n : int

The number of sides

Raises :

GeometryError :

If the \(center\) is not a Point, or the \(radius\) is not a number or Basic instance, or the number of sides, \(n\), is less than three.

Notes

A RegularPolygon can be instantiated with Polygon with the kwarg n.

Regular polygons are instantiated with a center, radius, number of sides and a rotation angle. Whereas the arguments of a Polygon are vertices, the vertices of the RegularPolygon must be obtained with the vertices method.

Examples

>>> from sympy.geometry import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r
RegularPolygon(Point(0, 0), 5, 3, 0)
>>> r.vertices[0]
Point(5, 0)

Attributes

vertices  
center  
radius  
rotation  
apothem  
interior_angle  
exterior_angle  
circumcircle  
incircle  
angles  
angles[source]

Returns a dictionary with keys, the vertices of the Polygon, and values, the interior angle at each vertex.

Examples

>>> from sympy import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r.angles
{Point(-5/2, -5*sqrt(3)/2): pi/3,
 Point(-5/2, 5*sqrt(3)/2): pi/3,
 Point(5, 0): pi/3}
apothem[source]

The inradius of the RegularPolygon.

The apothem/inradius is the radius of the inscribed circle.

Returns :apothem : number or instance of Basic

Examples

>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.apothem
sqrt(2)*r/2
area[source]

Returns the area.

Examples

>>> from sympy.geometry import RegularPolygon
>>> square = RegularPolygon((0, 0), 1, 4)
>>> square.area
2
>>> _ == square.length**2
True
args[source]

Returns the center point, the radius, the number of sides, and the orientation angle.

Examples

>>> from sympy import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r.args
(Point(0, 0), 5, 3, 0)
center[source]

The center of the RegularPolygon

This is also the center of the circumscribing circle.

Returns :center : Point

Examples

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.center
Point(0, 0)
centroid

The center of the RegularPolygon

This is also the center of the circumscribing circle.

Returns :center : Point

Examples

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.center
Point(0, 0)
circumcenter[source]

Alias for center.

Examples

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.circumcenter
Point(0, 0)
circumcircle[source]

The circumcircle of the RegularPolygon.

Returns :circumcircle : Circle

Examples

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.circumcircle
Circle(Point(0, 0), 4)
circumradius[source]

Alias for radius.

Examples

>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.circumradius
r
encloses_point(p)[source]

Return True if p is enclosed by (is inside of) self.

Parameters :p : Point
Returns :encloses_point : True, False or None

Notes

Being on the border of self is considered False.

The general Polygon.encloses_point method is called only if a point is not within or beyond the incircle or circumcircle, respectively.

Examples

>>> from sympy import RegularPolygon, S, Point, Symbol
>>> p = RegularPolygon((0, 0), 3, 4)
>>> p.encloses_point(Point(0, 0))
True
>>> r, R = p.inradius, p.circumradius
>>> p.encloses_point(Point((r + R)/2, 0))
True
>>> p.encloses_point(Point(R/2, R/2 + (R - r)/10))
False
>>> t = Symbol('t', real=True)
>>> p.encloses_point(p.arbitrary_point().subs(t, S.Half))
False
>>> p.encloses_point(Point(5, 5))
False
exterior_angle[source]

Measure of the exterior angles.

Returns :exterior_angle : number

Examples

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.exterior_angle
pi/4
incircle[source]

The incircle of the RegularPolygon.

Returns :incircle : Circle

Examples

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 7)
>>> rp.incircle
Circle(Point(0, 0), 4*cos(pi/7))
inradius[source]

Alias for apothem.

Examples

>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.inradius
sqrt(2)*r/2
interior_angle[source]

Measure of the interior angles.

Returns :interior_angle : number

Examples

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.interior_angle
3*pi/4
length[source]

Returns the length of the sides.

The half-length of the side and the apothem form two legs of a right triangle whose hypotenuse is the radius of the regular polygon.

Examples

>>> from sympy.geometry import RegularPolygon
>>> from sympy import sqrt
>>> s = square_in_unit_circle = RegularPolygon((0, 0), 1, 4)
>>> s.length
sqrt(2)
>>> sqrt((_/2)**2 + s.apothem**2) == s.radius
True
radius[source]

Radius of the RegularPolygon

This is also the radius of the circumscribing circle.

Returns :radius : number or instance of Basic

Examples

>>> from sympy import Symbol
>>> from sympy.geometry import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.radius
r
reflect(line)[source]

Override GeometryEntity.reflect since this is not made of only points.

>>> from sympy import RegularPolygon, Line
>>> RegularPolygon((0, 0), 1, 4).reflect(Line((0, 1), slope=-2))
RegularPolygon(Point(4/5, 2/5), -1, 4, acos(3/5))
rotate(angle, pt=None)[source]

Override GeometryEntity.rotate to first rotate the RegularPolygon about its center.

>>> from sympy import Point, RegularPolygon, Polygon, pi
>>> t = RegularPolygon(Point(1, 0), 1, 3)
>>> t.vertices[0] # vertex on x-axis
Point(2, 0)
>>> t.rotate(pi/2).vertices[0] # vertex on y axis now
Point(0, 2)

See also

rotation

spin
Rotates a RegularPolygon in place
rotation[source]

CCW angle by which the RegularPolygon is rotated

Returns :rotation : number or instance of Basic

Examples

>>> from sympy import pi
>>> from sympy.geometry import RegularPolygon, Point
>>> RegularPolygon(Point(0, 0), 3, 4, pi).rotation
pi
scale(x=1, y=1, pt=None)[source]

Override GeometryEntity.scale since it is the radius that must be scaled (if x == y) or else a new Polygon must be returned.

>>> from sympy import RegularPolygon

Symmetric scaling returns a RegularPolygon:

>>> RegularPolygon((0, 0), 1, 4).scale(2, 2)
RegularPolygon(Point(0, 0), 2, 4, 0)

Asymmetric scaling returns a kite as a Polygon:

>>> RegularPolygon((0, 0), 1, 4).scale(2, 1)
Polygon(Point(2, 0), Point(0, 1), Point(-2, 0), Point(0, -1))
spin(angle)[source]

Increment in place the virtual Polygon’s rotation by ccw angle.

See also: rotate method which moves the center.

>>> from sympy import Polygon, Point, pi
>>> r = Polygon(Point(0,0), 1, n=3)
>>> r.vertices[0]
Point(1, 0)
>>> r.spin(pi/6)
>>> r.vertices[0]
Point(sqrt(3)/2, 1/2)

See also

rotation

rotate
Creates a copy of the RegularPolygon rotated about a Point
vertices[source]

The vertices of the RegularPolygon.

Returns :

vertices : list

Each vertex is a Point.

Examples

>>> from sympy.geometry import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.vertices
[Point(5, 0), Point(0, 5), Point(-5, 0), Point(0, -5)]
class sympy.geometry.polygon.Triangle[source]

A polygon with three vertices and three sides.

Parameters :

points : sequence of Points

keyword: asa, sas, or sss to specify sides/angles of the triangle :

Raises :

GeometryError :

If the number of vertices is not equal to three, or one of the vertices is not a Point, or a valid keyword is not given.

Examples

>>> from sympy.geometry import Triangle, Point
>>> Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
Triangle(Point(0, 0), Point(4, 0), Point(4, 3))

Keywords sss, sas, or asa can be used to give the desired side lengths (in order) and interior angles (in degrees) that define the triangle:

>>> Triangle(sss=(3, 4, 5))
Triangle(Point(0, 0), Point(3, 0), Point(3, 4))
>>> Triangle(asa=(30, 1, 30))
Triangle(Point(0, 0), Point(1, 0), Point(1/2, sqrt(3)/6))
>>> Triangle(sas=(1, 45, 2))
Triangle(Point(0, 0), Point(2, 0), Point(sqrt(2)/2, sqrt(2)/2))

Attributes

vertices  
altitudes  
orthocenter  
circumcenter  
circumradius  
circumcircle  
inradius  
incircle  
medians  
medial  
altitudes[source]

The altitudes of the triangle.

An altitude of a triangle is a segment through a vertex, perpendicular to the opposite side, with length being the height of the vertex measured from the line containing the side.

Returns :

altitudes : dict

The dictionary consists of keys which are vertices and values which are Segments.

Examples

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.altitudes[p1]
Segment(Point(0, 0), Point(1/2, 1/2))
bisectors()[source]

The angle bisectors of the triangle.

An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half.

Returns :

bisectors : dict

Each key is a vertex (Point) and each value is the corresponding bisector (Segment).

Examples

>>> from sympy.geometry import Point, Triangle, Segment
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> from sympy import sqrt
>>> t.bisectors()[p2] == Segment(Point(0, sqrt(2) - 1), Point(1, 0))
True
circumcenter[source]

The circumcenter of the triangle

The circumcenter is the center of the circumcircle.

Returns :circumcenter : Point

Examples

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.circumcenter
Point(1/2, 1/2)
circumcircle[source]

The circle which passes through the three vertices of the triangle.

Returns :circumcircle : Circle

Examples

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.circumcircle
Circle(Point(1/2, 1/2), sqrt(2)/2)
circumradius[source]

The radius of the circumcircle of the triangle.

Returns :circumradius : number of Basic instance

Examples

>>> from sympy import Symbol
>>> from sympy.geometry import Point, Triangle
>>> a = Symbol('a')
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, a)
>>> t = Triangle(p1, p2, p3)
>>> t.circumradius
sqrt(a**2/4 + 1/4)
incenter[source]

The center of the incircle.

The incircle is the circle which lies inside the triangle and touches all three sides.

Returns :incenter : Point

Examples

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.incenter
Point(-sqrt(2)/2 + 1, -sqrt(2)/2 + 1)
incircle[source]

The incircle of the triangle.

The incircle is the circle which lies inside the triangle and touches all three sides.

Returns :incircle : Circle

Examples

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(2, 0), Point(0, 2)
>>> t = Triangle(p1, p2, p3)
>>> t.incircle
Circle(Point(-sqrt(2) + 2, -sqrt(2) + 2), -sqrt(2) + 2)
inradius[source]

The radius of the incircle.

Returns :inradius : number of Basic instance

Examples

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(4, 0), Point(0, 3)
>>> t = Triangle(p1, p2, p3)
>>> t.inradius
1
is_equilateral()[source]

Are all the sides the same length?

Returns :is_equilateral : boolean

Examples

>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t1.is_equilateral()
False
>>> from sympy import sqrt
>>> t2 = Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3)))
>>> t2.is_equilateral()
True
is_isosceles()[source]

Are two or more of the sides the same length?

Returns :is_isosceles : boolean

Examples

>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(2, 4))
>>> t1.is_isosceles()
True
is_right()[source]

Is the triangle right-angled.

Returns :is_right : boolean

Examples

>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t1.is_right()
True
is_scalene()[source]

Are all the sides of the triangle of different lengths?

Returns :is_scalene : boolean

Examples

>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(1, 4))
>>> t1.is_scalene()
True
is_similar(t1, t2)[source]

Is another triangle similar to this one.

Two triangles are similar if one can be uniformly scaled to the other.

Parameters :other: Triangle :
Returns :is_similar : boolean

Examples

>>> from sympy.geometry import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -3))
>>> t1.is_similar(t2)
True
>>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -4))
>>> t1.is_similar(t2)
False
medial[source]

The medial triangle of the triangle.

The triangle which is formed from the midpoints of the three sides.

Returns :medial : Triangle

Examples

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.medial
Triangle(Point(1/2, 0), Point(1/2, 1/2), Point(0, 1/2))
medians[source]

The medians of the triangle.

A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas.

Returns :

medians : dict

Each key is a vertex (Point) and each value is the median (Segment) at that point.

Examples

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.medians[p1]
Segment(Point(0, 0), Point(1/2, 1/2))
orthocenter[source]

The orthocenter of the triangle.

The orthocenter is the intersection of the altitudes of a triangle. It may lie inside, outside or on the triangle.

Returns :orthocenter : Point

Examples

>>> from sympy.geometry import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.orthocenter
Point(0, 0)
vertices[source]

The triangle’s vertices

Returns :

vertices : tuple

Each element in the tuple is a Point

Examples

>>> from sympy.geometry import Triangle, Point
>>> t = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t.vertices
(Point(0, 0), Point(4, 0), Point(4, 3))

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